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GATE Exam | Discrete Mathematics Propositional Logic One Shot | CS & IT — Transcript

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0:00I'll start from the logic because logic
0:02is very helpful. Logic will help you to
0:05build or it will try to understand what
0:08are the things that will be there. So
0:10let's start from that logic. See if I'll
0:12try to create the topics of logic.
0:16Either you can call this as the logic or
0:18you can also call as a mathematical
0:19logic. If I try to say that so the
0:22mathematical logic is basically divided
0:24into five to six topics. The first one
0:27which is called as a propositional
0:28logic.
0:35And second topic that we are having is
0:37called as a logical equivalence.
0:44And third topic we are having which is
0:46called as a inference rule.
0:51And uh now we are having which is called
0:53as a quantifier.
0:58And then we are having which is called
0:59as a nested quantifier.
1:07And then quantifier with inference rule.
1:15Quantifier with inference rule.
1:23This whole topic has been divided into
1:25the logic has been divided into six
1:27topics. Now if I will try to say that uh
1:31it has been divided into two models. The
1:33logic the first model is basically
1:35called as a basics of logic and second
1:37model is the advanced logic. So this is
1:39basically basic it will help you to
1:41understand to create the basics of uh
1:43the whole subject and once we are
1:45comfortable with that and out of that
1:47the most important one is logical uh
1:50sorry which is nothing but the inference
1:52rule because uh propositional logic you
1:55may have studied somewhere uh logical
1:57equivalence you may have also studied
1:59somewhere
2:01maybe in 11th or 12th you may have
2:03studied logical equivalence but the
2:04inference rule you may have not studied
2:06anywhere. So understanding the inference
2:08rule is very very important and now we
2:10are having this quantifier nested
2:12quantifier and then quantifier with
2:14inference rule. So this is very very
2:15important related to that. So basically
2:18the whole logic as I told you basically
2:20it will be divided into two models. The
2:22first model is basically the basics of
2:24logic and second model is advanced model
2:26which is a quantifier the nested
2:28quantifier and then quantifier within.
2:30This is basically the advanced model.
2:33Let's go and let's try to start why we
2:35requires the logic and what is the main
2:37important topic related to that. See
2:39many times when we are studying about
2:41the logic. So here I'm starting please
2:43everyone pay attention. See whenever we
2:45are starting the logic many times the
2:47students thinks that you know what is
2:50necessary to go and to understand about
2:52the logic because GATE is this
2:54examination GATE is very tough
2:56examination and why they will require
2:59the concept of a logic. Let me tell you
3:02see the logic will always help you to
3:06understand the validity of a statement.
3:08Logic will always help you to find out
3:11the validity of a statement. So for
3:13example suppose if few people have done
3:16or if you have clear gate examination
3:18and you are going for Mtech or you are
3:20going for MS program then at there you
3:23may write your research paper. So when
3:25you are writing your research paper you
3:27have to write the whole paper. So in
3:29that paper you will try to use English
3:31statement. So suppose if you have
3:33written any type of statements. So this
3:36statement is valid or not. So who takes
3:38the responsibility? So who will give
3:41this responsibility? So here the main
3:44concept is there which is nothing but
3:46the logic.
3:47So logic will help you to understand the
3:50validity of a statement. This statement
3:52is valid or not. Who will tell you this?
3:54The logic. So logic is very very
3:56important topic. You always try to
3:58understand this very very important
4:00mathematical logic will always help you
4:02to understand you know the check the
4:04validity of a statement now okay sir I
4:08understood that you're trying to say
4:09that the mathematical logic will always
4:12help us to understand the validity of a
4:15statement okay sir I understood but give
4:18me a broad perspective see if you want
4:20to understand the broad perspective so
4:23let's try to understand from a computer
4:25science point of view what is the main
4:26intention of a computer science point of
4:28view. They are thinking that they are
4:30knowing the programming language and
4:32then they will create a software. So
4:34this is the perspective of a computer
4:35science student and know the main
4:37perspective of a computer science
4:38students. They think that they they have
4:42the programming language and then they
4:44will write they will try to create the
4:46software right you will be having the
4:48programming language and then from
4:49programming language you'll try to
4:51create the software. But this is not
4:53just a computer science perspective.
4:54There are more than this. there is a
4:56very broad perspective. So in order to
4:58use the programming language you should
5:00have the algorithm and then you will try
5:02to implement this algorithm into PL and
5:05then from PL you'll try to have a
5:07software but in order to use the
5:10algorithm is basically a method okay
5:12programming language is like a syntax so
5:14you will try to use this method you will
5:16try to implement into programming
5:18language and from programming language
5:20you will be making your own software
5:23but algorithm we directly cannot go and
5:25cannot use it. uh for example suppose if
5:28I will tell you can you make one
5:30software which tells me by taking the
5:34matrix representation and which can tell
5:36me whether this graph is having a
5:38perfect matching or not right graph is
5:40having the perfect matching or not
5:41suppose for example suppose if you'll
5:43have a certain this type of software for
5:46example suppose if I will tell you can
5:48you make one software where your input
5:50should be any type of graph and your
5:53output should be uh whether it contains
5:55a perfect matching or not. So when you
5:58will try to use some graph algorithm and
6:00then suppose if you're using some graph
6:02algorithm then you should also require
6:04some kind of theorem. So always remember
6:06in algorithm somewhere you'll be
6:08requires a little bit of theorem. If you
6:10are making the software related to
6:12number theory you will always requires
6:14the theorem. If you're making a software
6:16related to mathematical tool you will
6:18also require the theorem. Whatever the
6:20type of softares you'll be making it you
6:22will require a little bit of theorems.
6:23So in whenever you'll try to make the
6:26algorithm algorithm does not directly go
6:28and directly use it basically uses the
6:31theorem it also implements theorem. But
6:34what exactly is a theorem? Theorem does
6:36not directly comes from one direct type
6:39of statements. Theorem basically comes
6:41from a you know theorem is basically
6:43comes from a conclusion. The set of
6:46statements which leads to the
6:47conclusion. So here there are different
6:49types of statements and then there is
6:51one type of conclusion here and then
6:54this conclusion will leads to what this
6:56conclusion will leads to theorem. So for
6:58example you will have the programming
7:00language from programming language
7:02you'll make a software and you will use
7:04that algorithm into programming language
7:06and programming language in order to use
7:08the algorithm you'll also requires the
7:10theorem but from where you are using the
7:12theorem theorem you're using where the
7:16multiple statements leads to conclusion
7:18and then this conclusion you're using as
7:20a theorem so sir basically this is the
7:23broad perspective related to your
7:24computer science but where is the
7:26mathematical logic okay I understood
7:28that you're saying that here the theorem
7:30mathematics has been involved the
7:31algorithm has been involved programming
7:33language is involved and you are trying
7:36to make a software so software is
7:38involved and you are running the
7:39software on the hardware so your digital
7:41logic COA operating system has been
7:44involved so this is basically the broad
7:45perspective of a computer science but
7:47where is this mathematical logic where
7:50you know exactly we're using this
7:51mathematical logic so now the
7:54mathematical logic it gives us power if
7:57Anybody will ask what exactly is a
7:59mathematical logic? Then you have to say
8:01that mathematical logic it will gives us
8:04power to check the validity of a
8:06statement. For example, suppose if there
8:08is one theorem you're trying to prove an
8:10theorem. So if you are writing this
8:12statement so this statement is valid or
8:14not this power is given by the
8:16mathematical logic. This statement is
8:18valid or not. This power is given by the
8:20mathematical logic. This statement is
8:22valid or not. This power is given by the
8:23mathematical logic. So I'm trying to
8:26tell you the whole perspective. If
8:28anybody will ask what exactly is a
8:29mathematical logic, you have to say that
8:32mathematical logic. It will gives us
8:34power to check the validity of a
8:36statement. Whether the statement is
8:38valid or not, it gives us power to check
8:40the validity of a statement. So this
8:42statement is valid or not. This
8:44statement is valid or not. So who gives
8:45this power? The mathematical logic will
8:48gives us power to check the validity of
8:49a statement. Okay. Now let's move ahead.
8:53So but sir from where you are using this
8:56type of statement. So there is one set
8:58of English statement. So in ninth or
9:0110th standard you may have studied the
9:03different type of English statement. So
9:05there are so many different types of
9:06English statement. There is a
9:08exclamatory statement, there is a
9:09questioning statement, there is ordering
9:11statements, there is imperative
9:12statements. So in eth or nth standard
9:16you may have your different type of
9:17English statement. So can I say that
9:19this is a set of all the English
9:21statement. Can I say that can I
9:23implement all the English statement into
9:25this type of statements? No, we cannot
9:27implement all the English type of
9:28statements into this type of statements.
9:31So we don't uses all the different types
9:33of English statements. We uses a very
9:35special set of English statements. Those
9:39set of statements are called as a
9:41factual statement. What is the meaning
9:43of a factual statement? They they they
9:46create a facts whether this is right or
9:48whether this is wrong. whether this is
9:50yes or whether this is no, whether this
9:52is true or whether this is false. So
9:54they creates basically the facts type of
9:56statement. Okay, they try to create the
9:59facts type of statements. So we only
10:02take those type of statements and we
10:04uses into the mathematics. So always
10:06remember in mathematics we do not
10:08dependence on every type of statements.
10:10We depends on a very factual statement.
10:13So what kind of factual statement?
10:15Factual statements are those type of
10:17statements which will have only two
10:18outcomes. Either yes or no. Either true
10:21or false. Either one or zero. So any
10:24statements which is having only two
10:26types of outcome. Yes, no. True, false,
10:28one, zero. Missile will go or not go. So
10:31this type of statements are called as a
10:33factual statement. So there are so many
10:35different type of English statement. For
10:36example, let's consider there is a
10:38questioning statement. If I'll if I'll
10:39tell you how are you? So we are not
10:41using this type of statements. We are
10:43only using the factual statements. So in
10:45mathematics this factual statements are
10:47called as a okay this factual statements
10:49are called as a propositional
10:50statements. What we are calling this as
10:53we are calling this as the propositional
10:54statement. So here we are having this
10:56and then that is called as a
10:58propositional statement. So now let's
11:00talk about what exactly is a
11:01propositional statement. So here we are
11:03carrying forward this. So now this is
11:06called as what? This is called as a
11:07propositional statement.
11:10So if anybody will be asking what
11:12exactly is a propositional statement you
11:14have to say that any type of statement
11:16any type of English statement which will
11:18have the two outcome whether we can say
11:20that yes or no or true or false one zero
11:26any type of statement which is having
11:27only two outcome then those type of
11:29statements are called as a propositional
11:31statement or which define some facts for
11:33example my name is Satish this is one
11:36type of factual statement whether this
11:38might be true or whether this might be
11:40false. This is a very simple statement
11:41which is called as a factual statement.
11:43Right? So this propositional statements
11:46are basically it is of two types. One
11:48type of statements are basically called
11:50as a simple propositional statement. I
11:53know that you may have studied this type
11:54of things but still I'll try to give you
11:56more insight about it. Please listen. So
11:59this propositional statement is
12:00basically of two types of statement. one
12:02is called as a simple propositional
12:04statements and then another one which is
12:06called as a compound propositional
12:08statement. So there are basically two
12:10different type of propositional
12:12statement. One is called as a simple
12:13propositional statement and another one
12:15is called as a compound propositional
12:17statement. So there are two different
12:18type of statements as we can see that
12:20here.
12:22So what is the difference between the
12:24simple and compound? If I will be
12:25talking about with respect to this then
12:27I can say that the simple propositional
12:30type of statements are basically those
12:32type of if I'm talking about simple
12:34propositional statements simple
12:35propositional statements are those type
12:37of statements which you cannot break it
12:40okay which is not breakable basically so
12:42simple propositional statements are
12:44those statements which are not breakable
12:46not breakable what is the meaning of not
12:48breakable for example if you will break
12:50it so there is no meaning into that so
12:53if you try to break it then it does not
12:54have any meaning. Okay, it does not have
12:56any meaning at all. And for example, my
12:59name is Satisho. So if you try to break
13:02it, then what will happen? It does not
13:03have any meaning. But compound
13:05propositional statements are those type
13:07of statements where you can break it
13:09into multiple simple propositional
13:10statement. For example, if I will write
13:13one this five is greater than three.
13:16See, this is one type of statement.
13:17Either this can be a true or it can be
13:19false. So five is greater than three. So
13:22if I'm writing this statement, this is a
13:24factual statement. So you can also
13:26consider this as the propositional
13:28statement. Why? Because it is defining
13:30either true or false. It is defining
13:32either yes or no. It is having only two
13:34outcome. So if I'm writing something,
13:36either this can be true or this can be
13:38false. For example, what is a compound
13:41propositional statement? Compound
13:42propositional statement is five is
13:44greater than or equal to three. So you
13:45can try to divide this compound
13:47propositional statement into two
13:48different type of two different type of
13:50simple propositional statement. One type
13:52of simple propositional statement you
13:53can write five is greater than three or
13:56here you can write which is nothing but
13:57five is equals to three. So if you'll
14:01try to understand visual I mean visual
14:03representation of a compound
14:04propositional statement. So compound
14:06propositional statement it is a type of
14:09simple prop it is a type of
14:11propositional statement where you can
14:13divide into multiple simple
14:14propositional. So this is your compound
14:17you can divide into two simple
14:19propositional statement. Now the doubt
14:21arises see sir it is necessary we have
14:24to divide into two simple propositional
14:26statement. So either S1 or S2 or we have
14:28to divide into multiple simple
14:30propositional statement. So there is a
14:32simple propositional statement this
14:33there is a simple propositional
14:34statement like this or there is another
14:36simple propositional statement. So see
14:39it is totally depends on the compound.
14:41Sometimes you can divide into two simple
14:42propositional statement. Sometimes you
14:44can divide into two or more. Sometimes
14:46you can divide into three. Sometimes you
14:48can divide into four. So it is totally
14:50based on your compound propositional
14:52statement. So for example here this is
14:54your compound propositional statement.
14:56So when you try to divide you divide at
15:00some point that is called as your
15:03division point. In division I mean
15:06division point in propositional logics
15:08are basically called as a connectives.
15:11Your statements are basically based on
15:13your connectives. So this is called as a
15:15connectives. So for example here see as
15:18you can see that this is basically your
15:19connective. So this is your whole
15:21compound propositional statement. Now
15:23you can divide into two simple
15:25propositional statement where the first
15:27simple propositional statement will be
15:28five is greater than three.
15:32Second propositional statement will be
15:33five if it is equals to three. So this
15:35is your compound propositional
15:36statement. That is your simple
15:37propositional statement. This is your
15:39compound propositional statement where
15:41this is your first simple. This is your
15:43second simple and here you will be
15:45having which is nothing but or. So this
15:48is called as a connective. We are having
15:50the different types of connectives but
15:52majorly we are talking about you know
15:54for example four major connectives here
15:57and the first one is your conjunction
16:02and then second one disjunction
16:07and third one is single implication
16:13and here you'll have a double
16:15implication.
16:22Okay. So you have basically four major
16:25different types of connectives. First
16:27one is a conjunction. Second one is a
16:28disjunction. Third one is a single
16:30implication. Fourth one is a double
16:31implication. Now okay. I hope that
16:33everybody has understood this slide.
16:35Please let me know that any doubt then
16:37I'll move on. So we are having the
16:39different types of connectives. Now
16:41let's talk about what exactly is the
16:43different types of connectives. The
16:44first one we are having this conjunction
16:47everybody knows about it. I'm just
16:48trying to give a little bit of brush up
16:50on this and then we'll start the actual
16:52logic conjunction.
16:54Now if I will be talking about this
16:56conjunction is basically defining as
16:58your and
16:59okay or the representation it is having
17:02this. So when to use this conjunction?
17:04So that is nothing but the most uh you
17:06know the important point when we will
17:08try to use this whenever you will see in
17:10English statement whenever you will see
17:11the and or whenever you will see the but
17:13then we can use for conjunction. So
17:17now the most important thing I'll try to
17:19explain you like this see there is one
17:22uh compound prepositional statement and
17:25then you'll have a one simple statement
17:26here and one simple statement too. Now I
17:30will I will tell you the summary of
17:32whole propositional logic and it's like
17:35this. We can only define the truth value
17:39directly truth value only for a simple
17:42propositional statement. We can we can
17:45directly define the truth value only for
17:48a simple propositional statement. So for
17:50example here this is a simple
17:52propositional statement. We can directly
17:55define the truth value of this. Directly
17:57define the truth value of this. This is
17:59another simple propositional statement.
18:00We can directly define the truth value
18:02of this. But if you will ask me can you
18:05directly define the truth value of this
18:07compound propositional statement then
18:09I'll say that no I cannot directly
18:11define the I cannot directly define the
18:14truth value of compound proposition
18:16statement. And why? Because see I do not
18:20have that authority to define the truth
18:22value. What exactly is a truth value?
18:24Truth value means you can define true or
18:26false. Right? You can find out the value
18:28of that statement. Truth value means
18:30you'll have two outcome either one zero,
18:32true, false, something like that. So
18:34again I'll I'll try to repeat it. I can
18:36directly define the truth value with
18:38respect to this statement only. I can
18:39directly define the truth value with
18:41respect to this statement only. I cannot
18:43directly define the truth value of this.
18:45So sir, how we can define the directly
18:47truth value of this? Directly we cannot
18:49do it. Indirectly we can do it. That
18:51means if I want to find out the truth
18:53value of this then I will ask to this
18:55okay tell me what is your value then the
18:58compound will say that I do not have any
19:00value go and ask to the simple
19:02propositional statement and that is why
19:04you will be having the you know the
19:06table so many times a student used to
19:09ask me sir what exactly is a table then
19:11I'll say that table is basically the
19:13case studies in order to find out the
19:16truth value of this compound so now if I
19:19want to ask what is your truth value
19:21then it will say that go and ask to the
19:23this type of simple proposition
19:25statement and that's the reason for
19:27example we are defining the variable for
19:29this and here this is the variable P
19:31which has been assigned to one simple
19:33proposition statement and there is
19:35another variable Q which has been
19:36defined for another simple propositional
19:38statement so for example here we are
19:41having this
19:43you know the truth table and I will say
19:46that this will become P and then this
19:47will become Q whenever you'll see and or
19:50but we have to use this conjunction and
19:51then that is nothing but P and Q. Now
19:54when the P and Q will or how we can
19:56define it everybody knows about it I'm
19:58just trying to giving a brush up a more
20:00inside story about it. So for example
20:02let's consider if this is true this is
20:04true then this statement will become
20:05true. For example this is true and then
20:07this is false then this statement will
20:09become false. If this is false this is
20:11true. So this statement will become
20:13false and if this is false this is false
20:16then this will become false. Now one
20:18thing you people have to understand very
20:20clearly and that is basically
20:26uh whenever if something is made up of
20:28your conjunction or whenever if
20:29something is made up of your and
20:30condition one thing you have to take
20:32care very easily and that is when a
20:36compound statement is made up of two
20:38simple prepositional statement and if
20:41one of them is false then it does not
20:43take care about this statement. what
20:45this statement is trying to say that
20:47this whole thing will become what? This
20:49whole thing will become false. So this
20:51is the major breakthrough that you can
20:53understand with respect to the
20:54conjunction. Suppose if there is a
20:56compound propositional statement and if
20:59it is made up of three simple
21:02prepositional statement and one of them
21:04is a false then what will happen? It
21:07does not matter what others are saying.
21:09Then it can say that the whole statement
21:12will become what? The whole statement
21:13will become false. then we can
21:15definitely say that the whole statement
21:16will become false. So suppose if there
21:19is one compound propositional statement
21:21and which is made up of four different
21:24type of simple propositional statement
21:26and one of them will become a false.
21:29Suppose even if it will become also
21:31false then it does not matter what what
21:34all others are saying then again the
21:36whole statement will become false. So
21:38that means the whole thing this whole
21:40thing will become false. So what you can
21:43have the assumption from this you can
21:45have a very simple assumption it says
21:47that key and hates false. So here I'm
21:50writing the statement and the statement
21:52you have to understand and hates false.
21:55Okay. So this is uh sometimes when we
21:58are reading about a quantifier this is
22:00the basic of the quantifier. In
22:02quantifier also I will try to teach you
22:04the same thing. See you know I know that
22:07I have a trust on you. you know
22:09everything but I'm trying to give you
22:10more inside story about it. Uh story
22:13like directly we cannot define the truth
22:15value or variable to compound
22:17propositional statement. That's
22:18someight.
22:20Second insight that and hates false. So
22:23this is the most important thing. See
22:25I'm not trying to tell you that you have
22:26to remember this table but I'm trying to
22:28tell you you have to remember this
22:30statement and that statement says very
22:32easily and it says that and hates false.
22:34So it's a very simple and it's a very
22:37easy concept that you people can try to
22:40understand. I hope that this has been
22:41clear to each and everyone. Please tell
22:43me that clear.
22:46Yes sir. Yes sir. Okay.
22:51Now we are having the second type of uh
22:53second type of you know the important
22:56thing and then that is called as a
22:57conjunction. Conjunction is basically
22:59your or condition and okay and we are
23:04having the two different types of or
23:06okay one is called as your exclusive or
23:09exclusive or exclusive or we are mostly
23:12learning into dist logic
23:14and second type of incl or is called as
23:17your inclusive or inclusive or is
23:19related to your mathematics. So what is
23:22the difference between exclusive or and
23:24inclusive or? See if I talking about
23:26exclusive or it says that one or
23:29another. One or other that's it one or
23:33other that is related to what? That is
23:34related to your exclusive or see if I'll
23:37talk about inclusive or that says that
23:39one other or both. One other or both. So
23:43this is related to what? This is related
23:45to your inclusive. So in mathematics we
23:48we learn more about inclusive or So
23:50let's try to take one example related to
23:52inclusive or. So for example here
23:54suppose if I'll be talking about your
23:56truth table. So let's consider this is
23:58your P this is Q and then here you'll
24:00have a P or Q. Suppose if this is true
24:02this is true then this will become true.
24:04If this is true this will false then
24:06still this statement will become true.
24:07If this is false this is true. Still
24:10this statement will become true. And if
24:11this is false this is false this
24:13statement will become false.
24:16What you can understand from this point
24:17is basically if at least one of the
24:20statement is true then the whole
24:22statement will become true. Now what we
24:24can have the assumptions related to
24:26this. Suppose if there is one statement
24:28which is made up of two simple
24:30propositional statement and if one of
24:32them is a true then what will happen? It
24:34does not matter what you are writing
24:36then the whole statement will become
24:37true. Second assumption or second uh
24:40criteria. For example, suppose if your
24:43compound propositional statement is made
24:45up of three simple propositional
24:47statement and if one of them will become
24:50a true then it does not matter what the
24:53others are saying then the whole
24:55statement will become what? Then the
24:56whole statement will become true. So
24:58here also if if your compound is made up
25:01of two simple prepositional statement
25:03and if one of them is a true then true
25:05will see the or then the whole statement
25:07will become true. Here if your compound
25:09is made up of three simple propositional
25:11statement and one of them is a true then
25:13true will see the or the whole statement
25:15is true. In a simple language we can say
25:17that or loves true. Okay. So these two
25:20things you have to remember. Again I'm
25:22not telling you to remember the truth
25:25table. Everybody knows about the truth
25:28table and here I can say that one or
25:30other or both. So for example here as
25:32you can see that this is one then again
25:35this this this statement will become
25:37true or other again this statement will
25:40become true or here the both then again
25:42this statement will become true. So we
25:44can say that P or Q will become true. So
25:47this is related to what this is related
25:48to your conjunction. So you know this is
25:51the concept which is related to your
25:52inclusive or. So inclusive or says that
25:55one or other or both. What you can learn
25:57from this topic is basically or true. I
26:00can tell you one example so that you can
26:02understand more specifically related to
26:05this. So we have understood the concept
26:08of a dominating set if you'll remember
26:11that. So dominating set says that either
26:14the vertex directly belong either the
26:16vertex directly belong
26:20directly belongs
26:23or or its adjacent will belong or its
26:28adjacent
26:29or its adjacent belongs right so here we
26:34are having this statement so I hope that
26:36everybody remember that now as you can
26:38see that this statement is this is your
26:41first simple type of propositional
26:43statement and this is your second simple
26:44type of propositional statement so this
26:46will become P this will become Q now you
26:49can understand the power of mathematical
26:51logics Suppose if somebody has given the
26:54domination number statement in in some
26:57research paper and if you're reading
26:58this domination number research paper
27:00then you are saying that the first
27:02definition domination number says that
27:04either the vertex directly belong or its
27:06adjacent will belong. So you'll try to
27:09prove it by the truth table. This is P
27:10this is Q and then this is P or Q. This
27:13is your statement. Okay this is P or Q
27:16your statement or this is your theorem
27:18which is nothing but P or Q. Now what
27:20will happen? Suppose if a vertex
27:22directly belong that means if this is
27:24true then still you can consider your
27:26statement is also true. For example if
27:29the adjacent will belong if the second
27:31statement is also true. Still you can
27:33say that your theorem is your statement
27:35is true. Right? Still you can say that
27:37your statement is true. For example
27:40suppose if the vertex directly belong
27:43and its adjacent will also belong then
27:45still you can say that your statement
27:46will become true. So in that matter this
27:48is the actual implementation. This is
27:50the actual
27:52you can say that uh the representation
27:55or the actual thing related to this
27:58understood everyone.
28:00I think in the main topic I have given
28:03that conjunction dissection. Okay, this
28:06is resumption. Sorry.
28:09Thank you for pointing it out.
28:15Okay.
28:16Yes sir.
28:19The next one is very very important and
28:21then that is related to a single
28:23implication. See what exactly is a
28:25single implication. See the whole gate
28:28questions the whole gate questions
28:31revolves around this operator. If you
28:34will understand the truth table of this
28:36operator then you can understand all and
28:39then this is related to what? This is
28:41related to your single implication.
28:44Single implication. Sometimes it is also
28:47called as a conditional statements,
28:51conditional statement. Either you can
28:53call as a single implication or you can
28:55also call as a conditional statements or
28:57you can also call as a single
28:59conditional statements.
29:01The whole gate questions revolves around
29:03this. Even the most of the theorems
29:06okay most of your theorems are also
29:09revolves around this. So
29:12the way you understand this
29:17if you will understand thoroughly or if
29:19you'll understand in a good way related
29:22to this single implication or
29:24conditional then you can consider the
29:2730% of your job is done you just have to
29:30go through a little things related to
29:33that. So here we will have the single
29:35implication. Either you can call this as
29:37a single implication or you can also
29:39call this as a conditional. Now what is
29:42the representation of this? The
29:44representation of this is something like
29:46that. The first thing that when we have
29:49to use a conditional statement, when we
29:50have to use a single implication, we
29:53have to use a single implication. We
29:55have to use this notation. See everyone
29:58try to understand. We have to use this
30:00notation. And when you will see
30:03something like that or either you will
30:06write like this or you can also try to
30:08write like this.
30:10When we have to use this notation, we
30:12have to use this notation. First thing
30:15if you will see any statement that
30:17contains something like this. If P then
30:19Q then you have to use this statement.
30:22Another one if P comma Q then you have
30:26to use that statement or
30:30Q if P then you have to use that
30:32statement or whenever you will see you
30:36have a different types of
30:38this you know whenever you will see this
30:40statement you do not have to do anything
30:42whenever you will see this statement you
30:44have to use P implies whenever you will
30:46see this statement if P comma Q then you
30:49have to use this statement here Q if P
30:51whenever you will See this statement you
30:53have to use P implies Q. I mean this
30:55statement Q when P or whenever you will
30:58see P implies Q
31:01P implies Q then you have to use this
31:03statement. You do not have to do
31:05anything. You just whenever you will see
31:08this whenever you will see this you just
31:11have to use this. Whenever you will see
31:14this you just have to use this. So if P
31:17then Q then you have to use P arrow Q.
31:20If P comma Q then you have to use you
31:24know this statement. Q if P then you
31:26have to use this Q and P then you have
31:29to use this P implies Q then you have to
31:32use this right.
31:36So you have to use this whenever you
31:38will see something like that
31:45or whenever you will see Q unless
31:47negation of P then again you have to use
31:50this. I will give you the different
31:52types of things. Okay. So there are
31:54different. So if P then Q, Q if P, Q
31:59when P or Q whenever P
32:04or Q whenever P whenever you will see
32:07this type of statements you have to go
32:09and you have to use this type of things
32:11which is nothing but P implies. So if P
32:14then Q whenever you will see any of
32:16these things you just have to use this.
32:18Now let's try to understand the truth
32:20table. Once you will understand the
32:22truth table, you know the major things
32:24are over and how we can try to use it.
32:27Let me take one example. First I will
32:30teach you with the help of one example
32:32and it will be very important. You know
32:34it will become very easy once you will
32:36understand with the help of that
32:37example. And your example is something
32:39like that. For example, if I'm writing
32:42okay, if perfect matching exist, okay,
32:45if perfect matching exist, okay, then
32:49number of vertices will become even then
32:52number of vertices will be even then
32:55number of vertices number of vertices
32:59will be even
33:01number of vertices will become even
33:03number of vertices will be even. So if
33:06perfect matching exist then number of
33:08vertices will become even. So let's
33:10consider this is nothing but your
33:12statement. Let's consider this is
33:14nothing but your statement. Okay. If
33:15perfect matching exist then number of
33:17vertices will become even. So as you can
33:20see that how how to identify these
33:22things. So as here you can see that it
33:25is starting from which statement? it is
33:27starting from if if so if any statement
33:31starting from if either you can go with
33:33respect to the first one or you can go
33:34with respect to the second one so I'm
33:36going with respect to the first one why
33:38because here as you can see that this is
33:40nothing but your first operator and then
33:44you'll have the then and then here
33:45you'll have the second operator so for
33:47example I can see that this is nothing
33:50but your P and then this is nothing but
33:52your Q right so let's try to understand
33:56related to this. So if perfect matching
33:59exists then number of vertices will
34:00become even. Right? Suppose if I want to
34:03make a truth table of this as I'm
34:05telling you understanding the truth
34:07table of this will always be helpful. It
34:10will solve most of our major problems
34:13related to your logic related to your
34:15mathematical logic. So what it says it
34:18says key if perfect matching exists then
34:20number of vertices will always become
34:22even right. So if perfect matching
34:24exists the number of vertices will
34:26become even. Now suppose how to you know
34:31draw the truth table of this please try
34:33to focus. For example suppose if you're
34:36a new mathematician and you have given
34:40this a brand new theorem to this whole
34:43world. Okay. Suppose like this theorem
34:46or any other theorem or something some
34:48different different types of theorem.
34:50You have given a very brand new theorem
34:53to this world.
34:56And what is that theorem? The theorem
34:58says that if perfect matching exist then
35:02the number of vertices will be even. You
35:04have given this theorem to this whole
35:06world. So suppose if you have given this
35:08theorem to this whole world then what
35:09will happen? The people will come and
35:11people will try to destroy the theorem.
35:13Right? That's how the mentality. So
35:17let's consider there is a four different
35:19type of people and then they are trying
35:21to come and then they're trying to
35:23destroy your theorem.
35:25So first person so what are the cases
35:27let's take all the cases your cases
35:31maybe in some graph perfect matching is
35:33existing and let's consider that because
35:36there with respect to any one statement
35:38you will have a two possibility first
35:40possibility perfect matching is also
35:42existing even number of vertxes are also
35:44there second possibility perfect
35:46matching is existing second even number
35:48of vertxes are not there third
35:50possibility where the perfect matching
35:52is not existing and even number of
35:55vertices will be there and then the last
35:57possibility where the perfect matching
35:59is not existing and then the even number
36:01of vertices is also not there. Let's
36:04consider you'll have a four different
36:05types of possibility. Let's consider the
36:08first condition perfect matching is
36:10existing. So here this is the first
36:12person and you have given a brand
36:15theorem and then there are four people
36:16there are four mathematician. So the
36:18first mathematician is coming and then
36:20first mathematician is coming with one
36:22example where he's saying that I'm
36:24having a one graph where the perfect
36:27matching is existing that means this is
36:29true. The first one is true and the even
36:32number of vertices are also getting
36:34true. So that means this person is
36:36saying that true implies true and then
36:39you'll be getting as a true right. This
36:41is the first person.
36:44So what this first person is doing? The
36:46first person is doing where the perfect
36:48matching is existing. So he is
36:50considering that the statement is true,
36:53right? He is considering that the
36:54statement is true. And then second point
36:57where the even number of vertices are
36:59also true. So true implies true. And
37:01your answer you are getting is basically
37:03true. Now let's talk about the second
37:05person. Now what will happen with
37:07respect to the second person? Second
37:10person says that there is a perfect
37:12there is a graph where the perfect
37:14matching is existing but the even number
37:16of vertices are not existing. See
37:18everyone is trying to destroy this
37:20theorem and I can bet on that this
37:24theorem will only be destroyed at this
37:26point only. That means if the first
37:29statement is true and the second
37:32statement is false then only the whole
37:34statement will become false. Then only
37:36your theorem is false. Why this is
37:38called as a conditional statement? Why
37:40it is called as a implication? Because
37:42you are telling to the whole world you
37:45come up with any example where if the
37:49perfect matching is existing because you
37:51you have given this theorem and you're
37:54you're telling to this whole world. You
37:56come up with any example where the
37:58perfect matching existing. If the first
38:00condition is existing then 100% the
38:03second condition will exist. You're
38:04giving a condition to each and everyone.
38:06That is why it is also called as a
38:07conditional statement. You're giving a
38:09condition to each and everyone. You're
38:11saying that if this condition met then
38:14the 100% the result will be like this.
38:16So you have promised this to everyone.
38:18Now what happened the second person
38:20second mathematician he satisfy the
38:23first condition but the here the second
38:25condition is not existing. You try to
38:27consider for example this is already a
38:29theorem. Okay everybody has proven this
38:31is theorem. Let's we are considering one
38:34example where the you know just
38:36hypothetical example where the first
38:38condition is existing first condition is
38:40getting true but the second condition is
38:42not getting true but the second
38:43condition is getting false. So that
38:45means if the first condition is going to
38:48be true and if the second condition is
38:50going to be false then only we can say
38:52that the whole statement is false
38:54otherwise we cannot say that the whole
38:56statement is false.
38:58You understood these two points please
39:00let me know. Let's talk about the third
39:01statement. What is this third statement?
39:04Third statement will give you more
39:05clarity related to it. So for example,
39:08suppose if you have given this
39:09statement. You have given one
39:11conditional statement and what this
39:13conditional statement is trying to say
39:15that if perfect matching exist.
39:19Okay. But here see this this is saying
39:22that this statement is false that means
39:25you have given one condition.
39:28You are you're telling to this whole
39:29world if perfect matching existing but
39:33this person is come up with one example
39:36where the perfect matching is not
39:38existing that means can I say that this
39:40person is coming with the wrong example
39:42yes you you know you you're you're
39:45promising first you show your perfect
39:48matching exist first you show your
39:50perfect matching exist then I will tell
39:53you number of vertices will become even
39:54but this person is coming coming up with
39:57wrong example
39:58So if anyone is coming up with the wrong
40:00example, can I say that my theorem is
40:02wrong? No, my theorem is not wrong, your
40:04example is wrong. So that is why if the
40:07if the perfect matching is not existing
40:10but for example suppose if the number of
40:12vertices are even. Okay, let me take the
40:14another example where here the perfect
40:16matching is not existing but the per
40:19even number of vertices are there. So I
40:21can say that my theorem is true. I
40:23cannot dis you know I cannot
40:26discard my theorem my theorem will be
40:28true. So that is why if the false
40:30implies true then only we can say that
40:33the whole statement is going to be in
40:35the same way suppose if this is also
40:37going to be false and if this is also
40:39going to be false. So that means if my
40:42first thing is going to be false then I
40:45can say that the whole statement is
40:46going to be true. Whole statement is
40:48going to be true means my theorem is
40:51true. I cannot say that my theorem is
40:54wrong. So this is the most important
40:56thing and out of that the whole summary
40:59is basically when the something is false
41:03then it does not matter what you are
41:05writing on this second point then the
41:08whole statement will become what the
41:09whole statement will become true. So
41:11anywhere we can say that the false will
41:13always imply if something is false.
41:17False implies anything will always
41:18become a true. Why it is happening? The
41:20false is implying that means if the
41:23false is implying it says that you are
41:25taking a wrong example. So if you're
41:28taking a wrong example then you cannot
41:30say that your statement I mean you
41:31cannot say that your statement is wrong.
41:35So if this is going to be false that
41:37means if this is the wrong example
41:40taking a wrong example cannot disprove
41:42your theorem. So this is the most
41:44important thing which is related to
41:46this. Understood everyone.
41:48No it means whenever your first
41:51statement is false then we do not have
41:54to worry about the theorem. Okay. I'll
41:56I'll try to explain you more related to
41:58this. For example here
42:02if G is planer
42:06if G is planer
42:09then E is less than or equal to 3 N
42:12minus 6 right I give this theorem to
42:16each and everyone what the student used
42:18to what student used to give me the
42:21answer right they are saying sir
42:24take K33
42:26and tell me sir for K3 33 it is not
42:29something like that bro
42:32I gave you this theorem and this theorem
42:34is saying that if G is a planer you're
42:37coming with one example where this will
42:40become what this is your false that
42:42means if the left hand side will become
42:44false still my theorem is true you
42:46cannot disprove my theorem
42:49based on the wrong example you cannot
42:52disprove my theorem based on the wrong
42:53example this is a wrong example with
42:56respect to this I'm telling you take
42:58make a planer graph then we'll talk
43:00about this then I will prove it E is
43:02less than or equal to 3 N minus
43:05this is my theorem and you are coming
43:07coming up with one example where the K33
43:09K3 is nonpler I'm trying to tell you
43:12take the first statement as a true then
43:14we'll talk about E is less than or equal
43:16to 3 N minus 6 or I'm I'm trying to tell
43:18you something like that
43:22I I'm saying that take G is a planer and
43:24then we'll talk about E is less than or
43:26equal to 3 N minus 6 But you're coming
43:29with one example where K33 is
43:30non-planer. You take any planer graph
43:33then I will show you E is less than or
43:34equal to 3 N minus. So if I'm confident
43:37about this okay then what will happen?
43:40The most important thing that you can
43:42understand is if you have come up with
43:45any planer graph but I'm unable to prove
43:49this E is less than or equal to 3 N
43:51minus 6 then I can say that my theorem
43:54is wrong. My theorem is only wrong at
43:56one point of time. If I have promise
43:59left side is true and I'm saying that
44:02right side is false then only this will
44:05become false. Then only it will become
44:06false. Otherwise it will never become
44:09false.
44:10If I'm taking the first statement as a
44:12true and then the second statement will
44:14become a false. If you'll have a
44:16condition like this then only you can
44:18say about with respect to false
44:19otherwise you cannot say like that.
44:23Okay. So there are three different
44:26versions related to your implications.
44:28The for example if I'm writing the
44:30perfect matching is existing then number
44:33of vertices will become even. So there
44:35are three different version. First one
44:38which is called as a converse
44:42converse
44:45or it is also called as a vice versa.
44:50I here you will have this inverse
44:54and the third one which is called as a
44:56contraositive.
44:59There are three different variations.
45:01Now today I will prove you why you know
45:04uh when I was teaching you the planarity
45:06at that point of time I told you key
45:08this one is equivalent to contraositive.
45:10Today I will show you for example
45:12perfect matching existing even number of
45:14vertices. Let's consider this is P and
45:16then it implies Q. It is called as a
45:19vice versa or it is also called as a
45:21converse. Now what exactly it will
45:24become? It will become Q implies P. What
45:26is the Q? Q if you will say that it says
45:29that if any graph is having even number
45:32of vertices, can we say that the perfect
45:34matching is existing?
45:36If any graph is having even number of
45:38vertices, can we say that the perfect
45:40matching is existing? No. That means we
45:43can say that P implies Q is not
45:47equivalent to Q implies P. We can easily
45:52say that the P implies Q is not
45:54equivalent to Q implies P. That is
45:57related to your first statement which is
45:58nothing but your inverse. If I'm talking
46:02about with respect to inverse, inverse
46:03says that negation of P will implies
46:06negation of Q. So you have to take the
46:09negation of the first side and then it
46:11will implies negation of Q. Now if I
46:13will be talking about with respect to
46:14this what is your negation of P?
46:16Negation of P means negation of perfect
46:18matching. So it says that if there is no
46:22perfect matching in any graph there
46:25would not be any possibility of even
46:27number of witnesses. Can I say that this
46:29is also same as the first one?
46:35Can I say that this is also same as the
46:36first one?
46:42What is your opinion on this? It says
46:44that key if there is no perfect matching
46:46in a graph then there would not be even
46:49number of vertices. Are bro wrong. See
46:51here is no perfect matching but I can
46:54show you the number of vertices are even
47:02getting my point or not?
47:06Yes sir.
47:08So that is why this statement is also
47:10going to be false. I mean this statement
47:12is also not same as this. So P implies Q
47:16is not same as negation of P will
47:18implies negation of Q. Let's talk about
47:20the contraositive. What the
47:21contraositive says that contraositive
47:24says that P implies Q is same as
47:28negation of Q will implies negation of
47:30P. But what exactly is a negation of Q?
47:33What exactly is a negation of Q?
47:35negation Q what is Q? Q is even that
47:39means if there is no even number of
47:41vertices then it implies there is no
47:44perfect matching.
47:46So if it does not have the even number
47:49of vertices does not have even number of
47:50vertices what is the meaning of that?
47:52That means it will have suppose if if it
47:55will have the odd number of vertices
47:57then there would not be any perfect
47:59matching. So there is no perfect match.
48:01This is true. So that is why we can say
48:03that this statement is going to be same
48:05as this. Understood? Yes or no? Yes sir.
48:10Can I say that the converse and inverse
48:12are same basically
48:21that is the same thing are basically
48:24same. Yes or no?
48:28I can also try to prove with the help of
48:30this truth table. Okay. Let me do it
48:33fast. So here you'll have this P. This
48:36is Q and then this will become P implies
48:38Q. This is negation of Q. This is
48:40negation of P. And let's consider this
48:43is negation of Q implies negation of P.
48:45Now here this is true. This is true.
48:48This is true. This is false. This is
48:50false. This is true. This is false. This
48:53is false. Now P implies this is true.
48:56False. This is true. And then this is
48:58true. Q negation of Q. This is false.
49:02This is true. This line true will become
49:05false. False will become true. And true
49:07here will become false. False will
49:09become true. Negation of P here you will
49:12get false. This you will also get false
49:16and false will become true. And then
49:18false will become true here. Now
49:20negation of Q will implies negation of
49:22P. Negation of Q will implies negation
49:24of P. This will become true. True
49:26implies false will become false. False
49:29implies anything will become true. True
49:31will implies anything will become true.
49:33So as you can see that in truth table
49:38two columns if the two columns are
49:41equivalent to each other then we can say
49:44they are they are logically equivalent
49:45then we can say that they are logically
49:47equivalent. So I don't think you'll have
49:49any problem to this. What I'm trying to
49:51tell you that suppose if A is okay
49:56suppose if A is logically equivalent to
50:00B suppose if your A is okay just try to
50:03understand suppose if your A is
50:06logically equivalent to B when your A is
50:09logically equivalent to B that means
50:12another way of understanding is
50:14basically this is your column A and then
50:16this is your column B then what will
50:19happen column A and then column B both
50:22will have the same value. Another way of
50:24writing key A and B will have will have
50:29will have
50:31same values. Can I write like this? They
50:34will have the same values. So A and B
50:36will have the same values. A will have
50:38the same in instance. Whatever the
50:41values of A, the values of B will be
50:43there. See for example, whatever the
50:45values of A. So suppose if the values of
50:48a is also true then the value of this is
50:50also true. Another way of writing key a
50:54and b will have a same behavior. A and b
50:57will have will have
51:00will have same behavior. A and b will
51:03have a same behavior. What is the
51:04meaning of a and b will have a same
51:06behavior. Same behavior means suppose
51:09whenever a is true b is true. Whenever a
51:11is false b is false. Whenever a is true
51:13b is true. Whenever a is false b is
51:15false. That's the meaning of same
51:17behavior. As you can also say that if
51:19this is a, a means this column that
51:22means P implies P. B means negation of Q
51:24will implies negation of P. Whenever
51:26this is true, this will become true.
51:28Whenever this is false, this is false.
51:29Whenever this is true, true. Whenever
51:31this is true, this is true. That means
51:33they are having the same behavior.
51:35A and B will have a same behavior. So
51:37whenever two columns are same or two
51:40things are having the same behavior then
51:42we can say that they are logically
51:44equivalent to each other in mathematics.
51:47Okay then we can easily say that they
51:50are logically equivalent to each other
51:52in mathematics which is very very
51:54simple.
51:57Try to read this question and try to
52:00tell me that answer related to this.
52:04I hope that everybody can read it
52:09right.
52:11Okay. Tell me the answer related to
52:13this. What is the answer for the first
52:16one? Contraositive. Contraositive.
52:20Okay.
52:27For the second one, inverse. inverse.
52:35Third one, contraositive.
52:43Fourth one, inverse. Inverse
52:49and e
52:51converse.
52:53Converse.
52:54Okay. I don't think you need any type of
52:56explanation. So shall I shall we move
52:58ahead? Yes sir. Yes sir. Option A.
53:04What it is trying to say that the
53:06converse of the inverse. So first try to
53:09divide
53:11option
53:12A. I'm explaining the option E E
53:16elephant. Okay. E E what it is trying to
53:19say that the contraositive first you
53:22have to try to divide it. It says that
53:24contraositity of P implies Q. So here
53:27this is P implies Q contraosity of P
53:30implies Q will become negation of Q will
53:32implies negation of P. Now it says that
53:35take a inverse of that. So let's go to
53:38inverse. Where is that inverse? Inverse
53:40says that whatever it has been written
53:43take negative of both. For example
53:45suppose if something is like this. For
53:48example if something is A implies B then
53:51negation of A will implies negation of
53:53B.
53:55This is called as the inverse in a
53:57simple term. So here
53:59negation of Q will implies negation of
54:01P. Inverse of this negation of Q will
54:04become negation of the first one will
54:06become Q. Negation of the second one
54:08will become this. So it will become Q
54:10implies P. So now what is Q implies P.
54:14So the inverse of the contraositive of P
54:16implies Q. This will become the converse
54:20of P implies Q. Understood?
54:30Clear.
54:31So can last question answer also be
54:34inverse because they are logically
54:36equal. No, it is converse. Now because
54:39if you will take with respect to this
54:41with respect to this, it is converse.
54:46Okay. Yes sir. The contraosity of P
54:49implies Q. Contraosity of this is this.
54:52And then you have to apply inverse. So
54:56inverse of this is this. So this will
54:59what is this of this? This is the
55:02converse of this. Now
55:08now there is a one more concept and then
55:10that concept is called as a double
55:11implication.
55:16Either you can call as a double
55:18implication or it is also called as a by
55:20condition.
55:27by conditional. So either you can call
55:29as a double implication or by
55:30conditional. By conditional is having
55:32some representation and that
55:34representation will be something like
55:35that. That means it is going from left
55:38to right also and then it is also coming
55:40from right to left. So left to right is
55:42also there, right to left is also there.
55:44Now what it says? So whenever you will
55:46have this P and then whenever you will
55:48have Q and then it says that P double
55:50implication Q that means if this is true
55:53this is true then only this will become
55:55true. If this is false this is false
55:57then only it will become true. So that
55:59means whenever you will have the die
56:00implication whenever you will have the
56:02by conditional it will be only true when
56:05both will have the same behavior. That
56:07means when both the true then only it
56:08will become true when both the false
56:10then only it will become otherwise it
56:12will not become a true. So let's take
56:14one example or when to use the by
56:16conditional. Whenever you will see any
56:18statement where it consists of if and
56:21only if then only you have to use this
56:23condition or whenever you will see
56:25something like that. So whenever you
56:27will see something like this if and only
56:29if then you have to use the by condition
56:31or whenever you will see if then only
56:33you have to use the by conditional. So
56:35by conditional you have to use when when
56:37you will have something like that. For
56:39example, suppose if it is true and then
56:41if it is false then the whole statement
56:43will become false. If this is false and
56:46if this is true then the whole statement
56:48will become false. So can I say another
56:51way of writing key in double implication
56:54what is the requirement of this? In
56:56double implication the requirement of
56:58this is basically it should have both
57:00same behavior. Can I say like that yes
57:02or no?
57:08Can I say it should have a same
57:10behavior?
57:16Okay, let's take one example so that you
57:18can understand in a better way. For
57:20example, when I was teaching you about
57:22the graph theory, there I taught you
57:24graph. So any graph is a ulerian. There
57:29you can see that I have used if and only
57:31if. There I have used the concept of if
57:35and only if degrees of all vertices are
57:37even. Degrees of all are even.
57:41All are even. When you will try to see
57:43that when degrees of all are even then
57:46when this will become true. When your
57:48graph is ulian then only all degrees are
57:51even. So when the first one is true then
57:53the second one is true. Why it is
57:55happening? Because it we have used if
57:57and only if. Suppose if this is going to
57:59be false. Suppose if your graph is not
58:01ilarian then 100% the degree will also
58:03not become even can I say here both the
58:07conditional by conditional
58:09understood not understood
58:12okay now one more thing you have to
58:14understand suppose if there is a a is
58:16logically equivalent to b and whenever
58:19if logically a is logically equivalent
58:21to b. So here first condition when your
58:24A is logically equivalent to B and this
58:27is your A this is your B. So suppose if
58:30A is logically equivalent to B that
58:32means both will have a same behavior and
58:35if I will use double implication here
58:37then what I will get. If suppose if this
58:39is true this is true you will get this
58:41is true. If suppose this is false this
58:44is false again you will get true. If
58:46suppose this is true this is true again
58:48you will get true. If suppose this is
58:50false, this is false again you will get
58:52true. So can I say that all the instance
58:55of a implication will become true. So if
58:59at any point if all the instance are you
59:02are getting true then that expression is
59:04called as what is tology. That
59:07expression is called as what? That
59:09expression is called as a tologic.
59:11Simple understood? Yes or no?
59:15Yes sir.
59:17So
59:19logically equivalent will have three
59:22different type of definition. First A
59:25logically equivalent B. Another way of
59:28saying that A and B are having same
59:31behavior
59:32are having are having same behavior
59:37are having same behavior. A and B are
59:39having a same behavior.
59:42Or the third one a double implication b
59:45is tologic.
59:47Is to logic.
59:50Can I say that all these three
59:52expressions are same? First expression
59:55is equivalent to the second expression.
59:57First statement is same as second.
59:59Second statement is same as third. Third
1:00:01one is
1:00:02is same as first. These all are
1:00:04equivalent to each other. Either you
1:00:06will write like this or you can also
1:00:09write like this. A double implication B
1:00:11is a tautology. What is the meaning of
1:00:13this?
1:00:14That means A and B are logically
1:00:16equivalent.
1:00:18That is why you are getting a tology.
1:00:19Now if A implication is a tology, that
1:00:23means you will get the same value for
1:00:24all. So if the third expression is
1:00:26there, then third expression will also
1:00:28imply a double implication B. What do
1:00:30you say?
1:00:39What do you say? Please tell me fast.
1:00:43Sir, if a implication b is a tautology,
1:00:45then if a is false, then b is also
1:00:47false. Is it possible?
1:00:50Yes. Take example.
1:00:53If your graph is if your graph is not
1:00:56ilarian
1:00:58if your graph is not ilarian what you
1:01:00can say about the degrees of all
1:01:02vertices are even
1:01:06see it's a very simple if your graph is
1:01:09not a graph when your graph is not a
1:01:11graph
1:01:14when your graph is not graph if it
1:01:16doesn't cover all some vertices are not
1:01:20no some vertices are not even that is
1:01:22why your graph is not illegraph. Right?
1:01:24So if your graph is not illegraph then
1:01:26we can say that our degrees of all
1:01:28vertices are even then again if this is
1:01:30false this is false.
1:01:34Yes sir. Clear. Clear. I hope that this
1:01:38has been clear to each and everyone.
1:01:42We are having the different types of
1:01:44tables. Okay. Let's discuss about all
1:01:47the different types of tables here.
1:01:54So for example suppose if if there is
1:01:57any table where the last entities
1:02:01everything is a true. If the last
1:02:03entities everything is a true this
1:02:05expression is called as a tology. This
1:02:08expression is called as a tology or
1:02:11sometimes it is also called as a valid
1:02:13expression or valid statement.
1:02:16Suppose if the last statement everything
1:02:19is a false. If suppose if this is false
1:02:22also this is false also this is false
1:02:24also this is false also this expression
1:02:27is called as a contradiction.
1:02:30Contradiction
1:02:32contraositive is different contra do not
1:02:34get confused. This is called as a
1:02:36contradiction.
1:02:39Contradiction
1:02:42contraositive is a different for example
1:02:45suppose if there is any expression where
1:02:48at least one is true.
1:02:51When suppose if there is any expression
1:02:53where at least one is true at least one
1:02:57true.
1:02:58Suppose if any expression where this one
1:03:00is at least one true any expression at
1:03:03least one true then this expression is
1:03:05called as a satisfiable expression.
1:03:09satisfiable
1:03:11expression try to understand this for
1:03:14example I'll show you
1:03:17see
1:03:21okay try to see this table here in this
1:03:24table the last expression here as you
1:03:27can see that at least one is true so can
1:03:29I say that P and Q is a satisfiable
1:03:31expression
1:03:33can I say that P and Q is a satisfiable
1:03:35expression yes or no yes sir
1:03:38Right? P and Q will satisfy expression.
1:03:41Did you understand what exactly is the
1:03:42meaning of satisfiable expression? Yes
1:03:44or no? Please let me know everyone.
1:03:47Yes sir.
1:03:49Right. Okay. Suppose if everything is a
1:03:52true that means if everything is a true
1:03:56that is called as a valid or that is
1:03:58called as a tautology
1:04:00or you can consider as a this is a
1:04:02school of boys where
1:04:05school of boys boy school everyone is
1:04:08boy true is a boy this is a school of
1:04:10girl this is a school where
1:04:13in that class at least one boy is there
1:04:16for example suppose
1:04:19If
1:04:20this is expression, this is expression
1:04:23where it is not a tology where it is not
1:04:26a contradiction. It is a mixture of all.
1:04:29Suppose if it is not a total logic, that
1:04:31means you will get at least one false.
1:04:33Suppose if it is not a contradiction,
1:04:35then at least you will get one true. So
1:04:37this expression is called as a
1:04:38contingency. What it is called as a
1:04:40contingency.
1:04:43Contingency. So here this is my first
1:04:45statement. Can I write? All
1:04:47contingencies are satisfiable. What is
1:04:49your opinion? All contingencies
1:04:53are satisfiable. All contingencies are
1:04:56satisfiable. What will be your opinion
1:04:58on this? All contingencies are
1:05:01satisfiable. Can I say it is true or
1:05:03what? True.
1:05:07This is true or false? What what is your
1:05:09opinion on this?
1:05:13Can I write all valids are satisfiables?
1:05:20All valids are satisfiables.
1:05:24This is true or false? All valids are
1:05:26satisfiable. So true.
1:05:30Can I write all satisfiables are
1:05:32contingency?
1:05:36All satisfiables are contingency. What
1:05:38is your opinion on this? No. No also no
1:05:44false all satisfiables are not
1:05:47contingency
1:05:49all contingencies are satisfiable all
1:05:51valids are also satisfiable. I hope that
1:05:53this slide is clear to each and
1:05:55everyone. See for example suppose if I
1:05:59will give you any expression and you
1:06:01tell me whether it is toology or not. A
1:06:04and if I'm giving any expression here
1:06:10you tell me whether this expression is a
1:06:12satisfiable uh tautology or not.
1:06:17Tell me tell me with the help of truth
1:06:19table or any type of method. Try to use
1:06:22any method you want to use and tell me
1:06:25whether it is a tology or not. This
1:06:28expression is a tology or not. What is
1:06:30the meaning of this expression? That
1:06:32means at the end you have to come up
1:06:33with one concept where everything will
1:06:36become true.
1:06:38I know that everybody has used the truth
1:06:40table. What you people may have done? I
1:06:42mean what was the question? Let me tell
1:06:43you the question. The question was
1:06:45nothing but you have to check whether
1:06:47this is a tautology or not. So what you
1:06:49may have done, you have taken A, you may
1:06:51have taken B, you have taken C. And then
1:06:53later you have done a implies b and then
1:06:57what you may have done a and and then a
1:07:00implies b and then later you may have
1:07:03find out b implies c and then you have
1:07:06written everything and at the end I have
1:07:09what I have asked is basically is
1:07:12nothing but this is a and and then a
1:07:15implies b and b implies c and then the
1:07:20whole expression it is implying
1:07:23Right? You're asking I'm asking whether
1:07:25this one is a tautology or not. So that
1:07:27means if all the expression this is
1:07:29coming as a true true then we can say
1:07:32that it is a topology. Right?
1:07:35Okay. I'll teach you the shortcut
1:07:37related to this. So it has been seen
1:07:39that this type of questions there are so
1:07:42many type of questions related to this
1:07:44type has been seen into GATE examination
1:07:47and uh I will teach you the shortcut and
1:07:50then shortcut is something like that.
1:07:53For example in order to teach you
1:07:55shortcut what I'll try to I'll try to
1:07:59show you the truth table of the
1:08:02implication. This is the truth table of
1:08:05implication and you tell me where it
1:08:08will become a false
1:08:10where it will become a false.
1:08:14Sir, if that whole bracket inside is
1:08:16true and the C is false. Right? If your
1:08:20left side is a true and if your right
1:08:23side is a false, this is the only case
1:08:26where it is becoming a false otherwise
1:08:28it is not becoming a false. Am I right
1:08:30everyone?
1:08:32When your left side is a true,
1:08:35when your left side is a true and your
1:08:37right side is a false.
1:08:40Okay. When your left side is a true and
1:08:43your right side is a false, then only it
1:08:45will become a false. Otherwise, it will
1:08:47not become a false. Otherwise, it will
1:08:49never become a false. So, what I'm
1:08:51trying to tell you for example, suppose
1:08:53this is nothing but my expression.
1:08:57Suppose
1:08:58if I will try to check on that condition
1:09:01only.
1:09:03If I will try to check on that
1:09:05condition.
1:09:06Suppose if I will consider this left
1:09:09side as a true and if I will consider
1:09:12the right side as a false. Suppose if I
1:09:15will check because this is left side
1:09:17this is right side. If I will check on
1:09:20that conditions only and if if it
1:09:24converts as a false then I can say that
1:09:26it is not a tautology but if it will
1:09:28deny then I can say that it is a
1:09:30tautology. Do you understand what I'm
1:09:32trying to tell you?
1:09:34Say yes or no then I will explain more
1:09:36detail.
1:09:40For example, see suppose if there is one
1:09:43person okay and if this person is
1:09:47honest, if there is one person and if
1:09:49this person is honest and if I will put
1:09:52the pressure on this person suppose if I
1:09:54will put a pressure on this person and
1:09:57if it changes then that's not the on
1:09:59honest person honest person will never
1:10:02change based on the scenario based on
1:10:05the situation. If suppose if this is the
1:10:08person and if you will force on this
1:10:11person I mean if you'll if the society
1:10:14you know put a pressure on this person
1:10:16and if this person is an honest person
1:10:18this person will never change what I'm
1:10:20trying to tell you if any expression if
1:10:24this expression is a tautology
1:10:27okay for example if this expression is a
1:10:29tautology whatever I will do it can I
1:10:33say that it will change no it will ology
1:10:36and you have also find out that it's a
1:10:38tautology.
1:10:40You have also find it out that it is a
1:10:42tautology.
1:10:43Now whatever I will do it will never
1:10:45change. But what will happen? It will
1:10:47change. So my point my whole point is
1:10:50try to understand this. My whole point
1:10:52is I will try to make I will try to make
1:11:00I will try to make this statement as
1:11:04false. This statement
1:11:08as false. I will try to make this
1:11:10statement as a false. Right? Suppose if
1:11:15it changes suppose if it changes
1:11:19as false
1:11:22then I can say that it's not a tology
1:11:24then I can say that it's not a tautology
1:11:28then I can say that it's not a tautology
1:11:31it's not a tautology but suppose if it
1:11:34deny
1:11:36if it deny denying of what suppose if it
1:11:38will deny it will not change suppose if
1:11:41it will deny then I I can say that it is
1:11:44a tautology
1:11:52suppose if it is a if it is denying then
1:11:54I can say that it is a total logic okay
1:11:56let's try to understand the same
1:11:58statement here so what this statement
1:12:01says that this is a and a implies b and
1:12:06b implies c and then the whole
1:12:09expression it is implying C so this
1:12:13method is limited only okay this method
1:12:16is limited so that is why I'm trying to
1:12:18first explain this method later for for
1:12:21other questions you'll have another
1:12:23method so for example what we have to do
1:12:26you have to check for a false when so
1:12:29this method is applicable whenever you
1:12:31will have any expression which contains
1:12:35the implication and they are asking for
1:12:38tology so this expression this this
1:12:40method is only for that. So what you
1:12:42have to do? You have to take the left
1:12:44side as a true, you have to make right
1:12:45side as a false. Now here I am trying to
1:12:48take the left side as a true and I'm
1:12:51trying to take the right side as a
1:12:53false. So if I'm trying to take the left
1:12:56side as a true and if I'm trying to take
1:12:58the right side as a false. So here I got
1:13:00the value of C as a false. So I got the
1:13:03value of C as a false. Now let's
1:13:06everyone try to understand
1:13:09if I want to make the left side as a
1:13:11true and it is made up of and a
1:13:12condition. So it says that that means
1:13:16this must also be true and then this
1:13:19must also be true and then this must
1:13:21also be a true. So that means if a is
1:13:24going to be true see I got the value of
1:13:26a is a true. So here the value of a I
1:13:29got is a true. Now everyone please
1:13:31focus. See this whole thing is a true. I
1:13:34got the value of a as a true. Now this
1:13:38whole bracket is demanding true and I
1:13:41got the value of a as a true. Then what
1:13:44should be the value of b? The whole
1:13:46bracket is demanding true. The whole
1:13:49bracket is demanding true. I got the
1:13:51value of a as a true. What should be the
1:13:53value of b? The value of b has to be
1:13:55true. So again I got the value of b as a
1:13:57true. Now and I got the value of B as a
1:14:01true but what is the value of C bro is
1:14:04false. So true implies false will become
1:14:07false. Now false will see and and hates
1:14:10false. What will happen? The whole lefty
1:14:13side will become false
1:14:16and if the left side is false and false
1:14:20implies anything the whole statement
1:14:22will become true. So I was trying to
1:14:25prove this as false but it turns out to
1:14:27be what? It turns out to be a true. Sir
1:14:31is this also called proof by
1:14:32contradiction?
1:14:34Yeah, you can consider like that.
1:14:36Basically it is a proof by
1:14:37contradiction.
1:14:41Okay. So here we'll have this. Try to
1:14:44see this question. I'll try to solve
1:14:46one. You'll try to solve by yourself.
1:14:48Okay. Uh okay. I want you to give a
1:14:51Okay. This is true and this is true and
1:14:53okay for example first you try to solve
1:14:56this once you'll become handy then I
1:14:58will give you more liberty to solve all
1:15:00the questions okay try to solve this
1:15:03first second one
1:15:06I'm talking about second one see what
1:15:08you have to do see it's a very simple
1:15:11you have to take the lefty side as a
1:15:13true and you have to take right side as
1:15:15a false I know that it is the second
1:15:17time you are not able to solve this but
1:15:20I will help you to solve this. See the
1:15:23main intention.
1:15:25Okay fine. The main intention to solve
1:15:28this type of question try to find out
1:15:31the values of variable. See here first I
1:15:35know that this is the first time believe
1:15:37me after solving four or five questions
1:15:39everyone will become professional into
1:15:41this. Now try to consider the left side
1:15:43as a true here I have taken this as this
1:15:45is whole left side this is whole right
1:15:47side. So the main intention is always
1:15:50try to find out the values of each and
1:15:52every element. So here this is true. So
1:15:55it contains a large amount of variable.
1:15:57I'm not going there. My right side is a
1:15:59false. This whole bracket I need false.
1:16:03Inside bracket you will have or
1:16:05condition. So what should be the value
1:16:07of Q and what should be the value of S?
1:16:12Both are false. Both are false. Both
1:16:14should be false. That means the value of
1:16:16Q should also be false. Value of S
1:16:19should also be false. That means the
1:16:22value of Q must also be false. And value
1:16:24of S must also be false. Now for
1:16:28example,
1:16:29what are the values we are getting? We
1:16:31got the value of Q and we got the value
1:16:33of S. So here this is your S. Now this
1:16:36whole bracket needs because see the this
1:16:39whole left side is a true that means and
1:16:42it is made up of and condition. So that
1:16:44means this bracket must also be true.
1:16:47This bracket must also be true and then
1:16:49this bracket must also be true. Now try
1:16:51to understand we got the value of S. So
1:16:54here the value of S is false and then
1:16:58the whole bracket is demanding true.
1:17:01What should be the value of R? False.
1:17:04False. False.
1:17:06So here as soon as you are getting the
1:17:08value write down as it is. So value of R
1:17:10is false. Where is R? R is here. So as
1:17:14soon as you're getting the value try to
1:17:15substitute it. This whole bracket is
1:17:18demanding true. What is the value of R
1:17:21is false. What should be the value of P
1:17:23then? True. True. True. The value of P
1:17:27must be true. So here I got the value of
1:17:29P as a true now. So value of P we got as
1:17:34a true. What is the value of Q? We
1:17:36already got it. So we don't need to find
1:17:38it out. True implies false. This will
1:17:41become false. false will see and and and
1:17:45hates false. So the whole lefty side
1:17:48will become false and false implies
1:17:50anything will become true. Hence second
1:17:52one is a tautology. Did you understand?
1:17:54Now
1:17:59so it is not a topology or it is a
1:18:01tautology now sorry it is a because it
1:18:04is denying so left the whole left side
1:18:08will become false. Okay, again I'll try
1:18:11to tell you this false. False false will
1:18:13see and so the whole left side will
1:18:16become false and false implies anything
1:18:21and then the whole statement will become
1:18:23what? The whole statement will become
1:18:24true. So it is a tautology.
1:18:27Did you
1:18:30uh do you want me to discuss the
1:18:32previous method or the method that I
1:18:35have taught you? Because that method is
1:18:37totally based on one concept and then
1:18:39that is related to your type one.
1:18:42Okay, that is exactly related to what
1:18:44that is exactly related to your type
1:18:46one. So there are basically six
1:18:49different types of method that I will
1:18:51teach you into your mathematical logic.
1:18:54So only one type we have covered and
1:18:57there are type two is there, type three
1:18:59is there, type four is there. So there
1:19:02are so many different types of things
1:19:04will be there. So let's talk about the
1:19:06other type. But before asking about the
1:19:08other type,
1:19:11before asking about the other types, you
1:19:13tell me whether did you understand
1:19:15everything related to type one? Okay. Uh
1:19:19is it visible everyone?
1:19:22Question is yesible. Uh yes. So uh in
1:19:26this they have we have to use a type one
1:19:29and uh see if the question they have
1:19:32given that you have to go and you have
1:19:33to use type once and obviously we are
1:19:36not able to if even if you will use the
1:19:39truth table then what will happen it
1:19:41will take a large amount of time. So
1:19:43even if you use the truth table with
1:19:45respect to P then it will take a large
1:19:47amount of time. So obviously we do not
1:19:49have to go and we do not have to use the
1:19:51truth table because if you'll be using a
1:19:52truth table then ultimately it will be a
1:19:55problematic thing that is why we will
1:19:57not be using
1:19:59more than three cases it becomes large.
1:20:02Yes. So if you'll see the answer so this
1:20:05one is going to be valid and this one is
1:20:06going to be valid and this is not toy.
1:20:09This is notology. I hope that everyone
1:20:11got the same answer. Yes sir.
1:20:15Yes sir.
1:20:17Right.
1:20:20And uh let's talk about this is our
1:20:22second type of
1:20:25uh I can say that second type of gate
1:20:28questions here we are having. So let's
1:20:31answer B
1:20:34with respect to this.
1:20:37Yes sir.
1:20:39Let's try to go and let's try to find it
1:20:41out which one of them is not a topology.
1:20:43They are seeing that. And here if you
1:20:46will see you can get a different type of
1:20:48cases. I mean this is your tautology.
1:20:50This is tology. This is valid.
1:20:54Uh if I will take this is true. This is
1:20:57false. This is also tology I think. So
1:21:00yes the answer is true and false. Yes
1:21:03this is this is answer. This is not a
1:21:06tautology and this will become your
1:21:08answer.
1:21:10If you want I can also solve it for you.
1:21:18Sir please start sir.
1:21:21Do you want me to solve?
1:21:24Yes sir. Yes sir. See for example if you
1:21:28will see this question. So in this
1:21:31question uh little bit uh you'll be
1:21:35having the cases has been expanded. So
1:21:38for example, if I'm writing this
1:21:40equation here and then I will write
1:21:42which is nothing but a implies c and
1:21:44then this whole thing is implying and
1:21:47then this will this one is negation of b
1:21:50and then negation of b that will implies
1:21:52which is nothing but a and c. So here we
1:21:54will be having some equations like that.
1:21:57Now let's try to understand. So for
1:21:59example as you can see that what is our
1:22:02main moto? Our main motto is we have to
1:22:04take the left side as a true and we have
1:22:06to make the right side as a false. Now
1:22:08if I will try to take this statement as
1:22:10a true here and if I will try to take
1:22:12this statement as a false here. Now it
1:22:14has to be clear that this must be true
1:22:17and then this must be false. Now as you
1:22:20can see that if for example suppose if
1:22:23you're starting from the left side then
1:22:25left side A implies C is true you are
1:22:28getting three cases. So I will not
1:22:30involve in anything that will be having
1:22:32a three cases. So this is the first
1:22:34thing you should understand. You should
1:22:36not involve at any point of time where
1:22:39it will be having a three cases
1:22:42because it will take a lot of time. So
1:22:43obviously I will not starting from this
1:22:45because a implies c it is true. For
1:22:48example, this is a this is c. So when
1:22:51this when the whole statement will
1:22:52become true when this is true this is
1:22:54true. When this is false this is true.
1:22:57when this is false and then this is
1:22:58false. So basically a implies c is
1:23:02basically true at three particular
1:23:04cases. That means both can be true. This
1:23:06is one particular case. I mean first one
1:23:08is a true, second one is a true, first
1:23:10one is a false, second one is a true.
1:23:12This is the second case and then the
1:23:13third case you are getting first one is
1:23:15a false and second one is a false. So a
1:23:17implies c is true. Basically it will be
1:23:19having a three cases. So obviously we
1:23:21are not going for we are not going to
1:23:23involved into that. Okay that will be
1:23:26for sure. we are not going to involve at
1:23:28any point of time with respect to this.
1:23:30So that is not a case if you'll see it
1:23:32directly. So here we can say that
1:23:34negation of B it is saying that it is
1:23:36going to be true. So definitely we can
1:23:38say that the value of B will become what
1:23:40value of so our main moto is to find out
1:23:43the values of any particular element. So
1:23:46immediately we can say that immediately
1:23:48the value of any particular element will
1:23:50become anything. Here we're getting the
1:23:53value of negation of B will become
1:23:54false. Now we have to come up with one.
1:23:59It is totally based on experience. Just
1:24:02try to understand. For example, we are
1:24:04only getting the values of B. I mean see
1:24:06negation of B we are getting this as
1:24:08true. So the value of B we are getting
1:24:10is basically false.
1:24:13A implies C is basically true. So this
1:24:15is A, this is C. This is true. That
1:24:18means this is true. This is false. This
1:24:20is true. And then this is false. And
1:24:22then this is false. So a implies c is
1:24:25true at the three cases. While here we
1:24:27are having this a and c. When the a and
1:24:30c will become false, a and c will become
1:24:32false at three cases. That means the
1:24:34first one can true second can false.
1:24:36Right? Or another one for example uh try
1:24:40to understand like this when your a can
1:24:42be false and c can also be false and
1:24:45when your a can be false and then c can
1:24:48also be true or the third case that
1:24:50you'll have a key both can be false. I
1:24:52mean which we have already seen it.
1:24:55Okay. The other case this first one can
1:24:58true and second can false. Now if you
1:25:00will if you will try to understand very
1:25:03clearly we can take this particular
1:25:05case. See either you can take this
1:25:08particular case or you can take this
1:25:09particular case. So for example if I
1:25:11will put the value of a as a false and
1:25:13if I will put the value of c as a false
1:25:16then the left side will become true.
1:25:17Right? As everybody can see that if
1:25:20you'll put the value of a as a false so
1:25:22it will become false. If you'll put the
1:25:24value of c as false so it will also
1:25:26become a false. So false imply false you
1:25:28will get this as true that means this
1:25:30this this thing will become true. Now
1:25:32here if you will see that a and c what
1:25:34will happen? Both will become both will
1:25:36become false. So if you will try to draw
1:25:39the truth table with respect to the
1:25:41truth table at the end you are getting
1:25:43this as false. So we just have to find
1:25:45it out with respect to only one
1:25:47particular case. If something is a false
1:25:49then that statement is not a tutology.
1:25:51So very specifically we can consider one
1:25:54particular case and with respect to one
1:25:56particular case we can take this.
1:25:58Understood?
1:26:07Did you understand what just now I
1:26:08explained you?
1:26:12See
1:26:14for example
1:26:16let's consider this is nothing but your
1:26:18truth table and if this is nothing but
1:26:20your truth table and suppose if you're
1:26:23getting the last answer is a false you
1:26:26tell me that this expression is a
1:26:28tautology or not a tology
1:26:33this expression is a tautology or this
1:26:36expression is not a tautology.
1:26:40Yes. So even if you will see a single
1:26:42false over here then what will happen?
1:26:44We can say that this expression is not a
1:26:46tology. This expression is not a tology.
1:26:50Then this will become very simple.
1:26:52Right? This expression is not a
1:26:53tautology. Now you'll try to understand
1:26:56this expression here. Again I will try
1:26:58to go and I will try to write down this
1:27:00with respect to this. See for example uh
1:27:03what we have to do we have to take the
1:27:06left side as a true and we have to take
1:27:07the right side as a false. So when you
1:27:10want to take the left side as a true and
1:27:11when you want to take the right side as
1:27:13a false. So you know we are taking the
1:27:16left side as a true we are taking the
1:27:18right side as a false. If you will go
1:27:20with respect to the left side a implies
1:27:23c it is true then it will have a three
1:27:25cases. When you are going with respect
1:27:27to the right side it is false then what
1:27:30will happen? This statement must become
1:27:32true and then this statement must become
1:27:34false. So is there any method where I
1:27:38can make my left side as a true and
1:27:39right side will become false. If your
1:27:41left side will become true and if your
1:27:43right side will become false then what
1:27:46will happen?
1:27:47Then what will happen then we can say
1:27:49that that means we can easily try to
1:27:52make the whole statement as a false. So
1:27:54for example suppose if I will you know
1:27:56with the help of little bit of
1:27:57experience if I will put the value of a
1:28:00as a false and c as a false then what
1:28:03will happen this left side will become
1:28:04true which implies and let's consider
1:28:07this is also true and then which implies
1:28:10for example the value of a as false and
1:28:12value of c as a false. Now what will
1:28:15happen false implies false will become
1:28:17true. True implies this is true implies
1:28:19false and false will become false. True
1:28:21implies will become false. So the whole
1:28:24expression will turns out to be a false.
1:28:26So for example, if you will put this is
1:28:29your A, this is your B, this is your C.
1:28:31This is the truth table. If this is a
1:28:34true table, if you will put the value of
1:28:35A as a false and if you'll put the value
1:28:37of C as a false, even with respect to
1:28:40this also, even with respect to this
1:28:43also and the negation of B is true,
1:28:46negation of B is true, that means B is
1:28:48false. So even if you will put with
1:28:50respect to suppose if you will take A as
1:28:52false, B as false, C as false. What you
1:28:54can see in this truth table see I'm not
1:28:57talking about all the rows. So do not
1:28:59get confused. I'm not talking about all
1:29:01the rows. I'm talking about only one
1:29:03specific row where A is false, B is
1:29:05false, C is false and then the hair
1:29:08answer you'll be getting is false. So
1:29:09this expression is definitely not
1:29:11automatic. So this is the main concept
1:29:13actually.
1:29:15This is the main concept where A is also
1:29:17false, B is also false, C is also false.
1:29:20Then definitely we can say that here we
1:29:22are not getting anything but here we are
1:29:24getting false. So that is why this
1:29:25expression will become noted automation.
1:29:27Clear?
1:29:30Yes sir.
1:29:32Okay. So in that way you have to
1:29:34understand. Okay.
1:29:38I hope that everybody can solve this.
1:29:40This I think uh P implies R Q implies R.
1:29:45In this everything is a tautology.
1:29:48Uh we have solved these two. Now if you
1:29:51have any doubt related to any particular
1:29:53things you can tell me. Okay. What about
1:29:56this? In this also I think everything is
1:29:58a topology. I mean this and somewhere
1:30:02this is map.
1:30:04Uh
1:30:08okay. This one and this one somehow
1:30:10questions are same. Last two.
1:30:18Okay. Everything is a total logic here.
1:30:20I think most of the questions
1:30:23clear. Yes sir. The right side of post
1:30:27number C.
1:30:30This one.
1:30:33Yes sir. This one right? Okay. So for
1:30:37example uh your question is something
1:30:39like that. This it's saying that P or
1:30:43and then Q or R
1:30:47this is one entity and then and negation
1:30:51of Q and this whole thing is implying
1:30:56which is nothing but P or R. Now try to
1:30:59take the values. So for example suppose
1:31:02if I will take the whole statement as a
1:31:04true and then this statement will become
1:31:06false. So when the right side will
1:31:08become false when your P is getting
1:31:11false here and when R is getting false.
1:31:14So the value of P you are getting this
1:31:16as false and then the value of R you are
1:31:19getting this as false. Now if this this
1:31:22must be true that means this box must
1:31:25also be true and then this box must also
1:31:27be true that means if you're getting
1:31:30negation of Q as a true then what should
1:31:32be the value of Q then the value of Q
1:31:34you are getting this as false. So here
1:31:37as you can see that I mean this negation
1:31:40of Q will also become a true. So the
1:31:41value of we got the value of P as R
1:31:44false R as false and Q as false. So
1:31:46every value we got it. Now try to
1:31:49substitute it. Then what will happen?
1:31:51The value of P will become false or
1:31:53value of Q is also false or the value of
1:31:56R is also false. So false are false are
1:31:59false. Now what will happen? Then this
1:32:01whole thing will become what? This whole
1:32:03thing will become false. And now the
1:32:05false will see the and condition. Then
1:32:07what will happen? The whole left side
1:32:10will become false. And if the whole left
1:32:12side will become false. And if false
1:32:15implies anything, then what will happen?
1:32:17Then the whole statement will become
1:32:19what? The whole statement will become
1:32:20true. Understood?
1:32:23You do not have to do anything. You just
1:32:25have to you know very polite way you
1:32:28have to write down the values and you
1:32:29just have to come up and solve. That's
1:32:32it.
1:32:34Now let's move ahead and let's try to
1:32:36find it out. See if you will see very
1:32:39carefully with respect to the type one
1:32:41what I have done I have used which is
1:32:43nothing but your implication right and
1:32:46so when to use the type one because type
1:32:48one is having a limitation so when to
1:32:50use the type one you have to use the
1:32:51type one whenever there is a implication
1:32:54and then they're asking for a question
1:32:56check whether this question is a
1:32:57tautology or not so we have to use the
1:33:00type one so you have to be very specific
1:33:02now you must be thinking sir what will
1:33:04happen if there is some
1:33:07What will happen if there is a instead
1:33:09of implication if there is a double
1:33:10implication something like that or
1:33:12instead of implication if there is a and
1:33:14condition then we we have to use this
1:33:16type instead of implication if suppose
1:33:19if there is a or condition is there so
1:33:21there are so many you know the different
1:33:23types of things which will be related to
1:33:26that what I'm trying to tell you is you
1:33:31simply whenever there is implication you
1:33:33have to use type one that's the main
1:33:35part that's the main part whenever there
1:33:37is a implication you have to use the
1:33:39type that's the main part now suppose if
1:33:41there is no implication then what are
1:33:43the things that we have to go and we
1:33:44have to use it so for example if there
1:33:46is no if there is a if there is no
1:33:48implication suppose you can use the and
1:33:50condition or what will happen so in with
1:33:54respect to that we will be having the
1:33:56another type and then that type is
1:33:58called as a type two and then type two
1:34:00is called as a logical equivalence
1:34:03logical equivalence
1:34:07Logical equivalence. Logical
1:34:09equivalence. Logical equivalence is
1:34:11having a two different type of
1:34:13representation.
1:34:14This is your representation or sometimes
1:34:16it is also use the double implication.
1:34:18So in some cases in some questions
1:34:22either you can say that the double
1:34:24implication is also used and the three
1:34:28line is also used for logical
1:34:30equivalence. So yesterday I have taught
1:34:32you two different things, two or three
1:34:34different things. For example, when your
1:34:37A is logically equivalent to B, this A
1:34:40is logically equivalent to B. When when
1:34:43A and B are having same behavior, when A
1:34:47and B are having are having Okay, when A
1:34:50and B are having the same behavior,
1:34:53right? Same behavior. Whenever you will
1:34:55have something like that when A and B
1:34:57are having the same behavior then you
1:34:59can go and you can try to use this
1:35:01concept where A is logically equivalent
1:35:03to B where A and B are having a same
1:35:05behavior. This is the first point you
1:35:07have to understand A and B are having a
1:35:09same behavior. What is the meaning of
1:35:10it? That means A can be true and B can
1:35:13also be true. Suppose if A is a false B
1:35:15is also false. Another way of saying
1:35:17that suppose if your A is logically
1:35:20equivalent to I mean A double
1:35:22implication B is a tology
1:35:25a double implication B is a tology then
1:35:27we can say that you know this is a point
1:35:30number two so point number one is
1:35:32basically same as a point number two you
1:35:34have to remember this point number one
1:35:36is always the same as point number two
1:35:38point number A says that A is logically
1:35:40equivalent to B where A and B are having
1:35:42a same behavior what I'm trying to tell
1:35:44you here is basically
1:35:47what kind of type two questions you will
1:35:49get in type two they will give you
1:35:53this expression and then they will say
1:35:56that check whether this one is a
1:35:57tautology or not. So instead of using
1:36:01just a single implication they will try
1:36:03to give you double implication and then
1:36:06they will try to find it out to check
1:36:08whether this expression and this
1:36:10expression is a tology. So this is
1:36:13related to your type two. Suppose if
1:36:16they will give something like that where
1:36:18the first expression and then double
1:36:20implication second expression they are
1:36:22saying that check whether it is a
1:36:24tautology or not. Then how we will say
1:36:26we will say that this is your expression
1:36:29A and this is your expression B. So
1:36:31suppose if expression A double
1:36:33implication expression B is a tology
1:36:36they are asking suppose if they're
1:36:37asking another way of writing key you if
1:36:41you can find it out if they are
1:36:43logically equivalent then either you can
1:36:46say like that or you you can say like
1:36:47that now what is the meaning of
1:36:49logically equivalent logically
1:36:51equivalent again I'm repeating please
1:36:53focus in type two what kind of questions
1:36:56you can get you will have one expression
1:36:59double implication you will have second
1:37:01expression. So here you will have
1:37:03expression A double implication you will
1:37:05have expression B. So expression A
1:37:07double implication expression B is a
1:37:09tology. Suppose if they're asking some
1:37:11questions like this. Now suppose if they
1:37:14are asking some question like this then
1:37:16another way of understanding we can say
1:37:18that suppose if this expression is
1:37:20logically equivalent to this expression.
1:37:22Now what is the meaning of logically
1:37:24equivalent? Logically equivalent means
1:37:27whenever they are having the same
1:37:28behavior. That means suppose if any
1:37:30cases if this whole expression is a true
1:37:33then this must also be true. Suppose if
1:37:36this whole expression is a false then
1:37:38this statement will also be false. So
1:37:41this is another another way of proving
1:37:43it or there is one more method and that
1:37:46method is we have to solve a and we have
1:37:49to solve b and we have to make a look
1:37:52like like b or we have to make b looks
1:37:55like like a. So how we can do that? We
1:37:58are having the different types of
1:37:59formulas and then then we can go and we
1:38:01can try to use it. I know that whatever
1:38:04I'm teaching you are not getting it
1:38:05until unless we'll solve some kind of
1:38:07questions. So instead of solving those
1:38:11kind of questions first we will try to
1:38:13understand some formula. So that formula
1:38:15will help you to make a looks like like
1:38:19b or you can make the b looks like like
1:38:22a that is the formula. Let's try to
1:38:24understand the type two today and that
1:38:26is related to logical equivalence. So
1:38:28for example here these all are the
1:38:30methods. First suppose if I'm having a
1:38:33and a then I can say that it is
1:38:35logically equivalent to a. Suppose if
1:38:37I'm writing a or a then it is also
1:38:39logically equivalent to a and then
1:38:41everybody knows about it right second
1:38:44type of formula suppose if I'm writing a
1:38:47and two and then I should be having some
1:38:49value and I don't know key what is a and
1:38:52true will looks like then what I'll say
1:38:54that suppose if this is a a and true now
1:38:57what I can say that suppose suppose
1:39:00suppose if I I don't know what is the
1:39:03value of this or what exactly the a
1:39:05entry a and true will looks like. So
1:39:08what I'll say that I'll try to
1:39:09substitute the value of a. Suppose if
1:39:11I'll substitute the value of a as a true
1:39:14then what I will get? I will get the
1:39:16answer as a true. So if you are
1:39:18substituting the a as a true you are
1:39:20getting answer as a true. Suppose if
1:39:22you're substituting the a as a false
1:39:24then what you will get the answer you'll
1:39:26be getting is basically false.
1:39:29So what I'm trying to tell you here
1:39:31whatever the values of a you're
1:39:33substituting you're getting the same
1:39:34value. So here a and true my answer will
1:39:37become a. Now suppose if I'm writing a
1:39:40or false then what is the value of you
1:39:42know this equation. For example if I'm
1:39:45writing a or false. So the value of a
1:39:48can be either true or it can be false.
1:39:50For example suppose if you put the value
1:39:52of a as a true then true or false will
1:39:55become true. Suppose if you're putting
1:39:56the value of a as a false then false or
1:39:59false you're getting false. That means
1:40:01whatever the values of a I will put it I
1:40:03will always be getting the same answer.
1:40:06So this is the second type of you can
1:40:09try to understand this is not related to
1:40:11type one basically it is related to uh
1:40:15basically it is related to what
1:40:17basically it is related to your uh you
1:40:19can say that you know logical
1:40:21equivalent. So a and true the answer we
1:40:24are getting as a and a or false you're
1:40:27getting answer as a. Now let's talk
1:40:30about the third one. Suppose if I will
1:40:32be writing a or true and then what
1:40:34should be the answer. I don't know the
1:40:36values of anything here. Then a or true
1:40:40is there. Right? So everybody knows that
1:40:43or loves true. So you know easily we can
1:40:46say that or looks true here. Right? So
1:40:50the whole answer will become true. Now
1:40:52if I will be writing a and false then
1:40:55what will happen? And hates false. And
1:40:57everybody knows that this equation where
1:41:00and hates false. So I can easily try to
1:41:03say that the whole statement will become
1:41:04what? The whole statement will become
1:41:06false. So this is nothing but related to
1:41:07this. Now let's move. These all are the
1:41:10things which is very much important when
1:41:13you want when you will have a very large
1:41:15expression a and when you will have a
1:41:17very large expression b. So whenever
1:41:19you'll have expression A whenever you'll
1:41:21have expression B so you try to solve
1:41:24expression A and then you'll try to
1:41:26looks like like B or whenever you'll
1:41:28have expression B you'll try to solve B
1:41:31and you'll try to make a looks like like
1:41:33A. So in order to do that we will
1:41:36require the different kinds of formulas
1:41:38here. Now let's move to the fourth one.
1:41:40Here I can say that key A or B is
1:41:43logically equivalent to B or A and A and
1:41:46B and then this is also logically
1:41:48equivalent to B and A. See
1:41:52this looks like very simple right? A and
1:41:55B is logically equivalent to B and A and
1:41:58A or B is logically equivalent to B or
1:42:00A. This looks like very simple but when
1:42:04the large amount of things you know it
1:42:07comes in your practical point of view
1:42:10then this will all this is very
1:42:13important to solve. For example suppose
1:42:15if you'll have a P or some expression
1:42:17something like that and then this is A
1:42:20and B implies R. Let's consider this is
1:42:22nothing but your one expression. Okay
1:42:25let's consider this is nothing but your
1:42:26one expression. Now what will happen?
1:42:28This expression will something like like
1:42:30this. Then this we can also in another
1:42:33way we can write a implies and then b
1:42:36implies r. This is your one box and then
1:42:39you can write or p. Now what we are very
1:42:43much handy whenever we will have this
1:42:46expression then we can easily try to
1:42:48write down this expression. Whenever we
1:42:51will have this expression then we can
1:42:52write down this expression. Right? This
1:42:54is where we are very much handy. But my
1:42:57main point is whenever you will have
1:42:59this expression then we are not getting
1:43:02sometimes it's not that easy to get
1:43:04click such that we can also write p or
1:43:07and then a and then b implies r. So do
1:43:10not think that this is just a or this is
1:43:14just a b. This a you can consider as a
1:43:17compound propositional statement. This b
1:43:19also you can consider as a prop compound
1:43:21propositional statement. Do not think
1:43:23that a is your simple propositional
1:43:25statement. Do not think that V is your
1:43:27simple propositional statement. You try
1:43:29to think like A as a compound
1:43:30propositional statement or in another
1:43:32way you can say that your A will be
1:43:34looks like like this and then your B
1:43:36will be looks like like this. So here
1:43:39this is your A and then this is your B.
1:43:41You'll try to consider like this. Now in
1:43:43A you can put as many as variables you
1:43:46can you know put it in B you can put as
1:43:49many as variables you can put it. So
1:43:51that's how you can try to get it. Okay.
1:43:54So this is related to fourth point where
1:43:57A and B they are not basically the
1:43:59simple propositional statement but
1:44:00basically they are the compound
1:44:02propositional statement. So you always
1:44:04try to you know understand related to
1:44:07this. So this is very very important
1:44:09whenever you will try to understand.
1:44:11Okay. So that's the main point.
1:44:15So let's move to the another you know
1:44:19the relations. So suppose if you'll move
1:44:21to the fifth point then here we will be
1:44:23having which we can write as a or and
1:44:26then b or c and then this is logically
1:44:29equivalent to a or b and then or c and
1:44:33another way of writing key a and and
1:44:36then b and c this is also logically
1:44:38equivalent to here a and b and then and
1:44:42c right so whenever you'll be having
1:44:44related to the fifth point again you do
1:44:46not have to am I audible
1:44:51Right?
1:44:52So do not think like that. A is a
1:44:55different simple propositional
1:44:56statement. B is a simple propositional
1:44:58statement.
1:45:04Try to think as a A is a compound
1:45:05propositional statement. B is also
1:45:07compound propositional statement. These
1:45:09all are simple. Till now I don't think
1:45:11we'll have any type of problem related
1:45:12to your first one, related to second,
1:45:14third, fourth and fifth. Now suppose if
1:45:16we'll move to the sixth one here the
1:45:18sixth one is very very important so
1:45:20please pay attention for example if I
1:45:22will write a or and then b and c see
1:45:27this is called as a commitative law
1:45:29every law is having a different
1:45:30different types of name so do not think
1:45:33like that you'll have the different
1:45:35types of name or something like that I I
1:45:38don't want see I do not believe on this
1:45:40you have to remember each and every name
1:45:42there are different types of name for
1:45:44example important is there. Okay. And
1:45:46then this is identity law and then this
1:45:49is uh you can say that you know the true
1:45:53law and uh this one you can say
1:45:56commutative law this one is associative
1:45:58law and this one is a distributive law.
1:46:01I'm not telling you to remember any
1:46:02name. I'm just trying to tell you based
1:46:06on the operator you have to play based
1:46:08on the operator.
1:46:12I will say that even in examination if
1:46:14you because if I will force you to
1:46:16remember the different types of name
1:46:19100% you will you know you you're going
1:46:21to forget it
1:46:23and this type of law are used in a
1:46:26different different places in d logic
1:46:29boolean algebra also you'll have this
1:46:30kind of law in set theory also you'll
1:46:32have a different this same kind of law
1:46:35and here also you'll have a same kind of
1:46:37law so I'll say that you do not try to
1:46:39use okay do not try to use any different
1:46:43type of name what I can say that you try
1:46:46to see the operator and based on the
1:46:47operator you'll take your decision so
1:46:50here this is a fifth one so how you can
1:46:53identify the fifth one your operators
1:46:55are same basically so a or b or c it's
1:46:59same as a or b or c a and b and c is
1:47:02basically same as a and b and c so your
1:47:05operators are same so whenever your
1:47:07operators are same you can switch the
1:47:10bracket
1:47:12Whenever whenever your operators are
1:47:14same you can switch the bracket. So for
1:47:16example here as you can see that a or b
1:47:18or c the bracket is here that means with
1:47:21respect to this the first you have to
1:47:23solve this particular bracket. Okay. And
1:47:26then in this the first you have to solve
1:47:30this particular bracket. Again I'm
1:47:31forcing on this particular point this a
1:47:33b and c they are not just a simple
1:47:37propositional statement. Basically they
1:47:39are a compound propositional statement.
1:47:41So always try to remember this which is
1:47:43nothing but a b and c.
1:47:46Okay. Anyhow let's move to the sixth
1:47:48point. The sixth point how to identify
1:47:50the sixth point. In sixth point your
1:47:52operators are different. Okay. Try to
1:47:55understand that your operators are
1:47:56different. So here a or b and c and then
1:47:59it is logically equivalent to we can
1:48:01write which is nothing but a or b and
1:48:03here this is and and then this you can
1:48:06write as a a or c. So this is called as
1:48:08a distributive law and everybody knows
1:48:10about it. Now here I can write this as a
1:48:12and and then this will become b or c and
1:48:16again you can say that your operators
1:48:18are different here your operators are
1:48:20different a or b and c. So these
1:48:23operators are different a and b or c
1:48:26again your operators are different here.
1:48:29So this is taking a entry and here it
1:48:32will become a and b and then this will
1:48:34become or and then this will become a
1:48:37and c. So this is the most important one
1:48:41distributive. Now I know that you may
1:48:43have studied this somewhere in 11 12 I'm
1:48:46just doing a brush up related to this.
1:48:48My point is very simple.
1:48:51Many times we just remember the formula
1:48:53and then that is moving from the left
1:48:55side to right side. What is the meaning
1:48:57of moving from left side to right side?
1:49:00Meaning of moving from left side to
1:49:02right side is basically I can write A or
1:49:05B and C. For example, suppose if you'll
1:49:08have something like that A or B and C.
1:49:10Then
1:49:12if this
1:49:14you will have this this is your left
1:49:16side and then when you will go to the
1:49:18right side what you will get? You'll get
1:49:20A or B and then here and it will become
1:49:23A or C. Right? So this will take a entry
1:49:26here A or B and C. You will get A or B
1:49:29and you will get A or C. So you will
1:49:32have the left side and you'll try to
1:49:33take a entry and you'll try to make the
1:49:36right side. A or B and A or C. My point
1:49:40is very clear in see sometimes your left
1:49:43side has not been given only the right
1:49:45side is given. So you have to identify
1:49:49your right side and you try to make that
1:49:52right side as a left side. Okay. So how
1:49:55you will try to identify that's the main
1:49:57problem. So for example let's consider
1:50:00this is a or a or and this is also a or
1:50:04a. So you'll be having in the left turn
1:50:06also you'll have a a or a in the right
1:50:09side also you'll have a a or a try to
1:50:12have a string on this. Try to say that
1:50:15try to consider a R. You know try to
1:50:19consider like this. Try to plug one
1:50:21string on a R and try to plug one string
1:50:23on A and then plug plug it and then you
1:50:27know remove it like that. Then what
1:50:28you'll get? You'll get the left side
1:50:30which is nothing but B and C. So
1:50:32sometimes it is very much important
1:50:34related to this type of clear.
1:50:39Yes sir.
1:50:42Right. Good.
1:50:45So this is a distributive law and then
1:50:47this is very important. Now let's move
1:50:49to the next type of law and then that is
1:50:51called as absorption. So for example
1:50:53here if I will write a or and then a and
1:50:57b and then this whole thing will become
1:51:00a or another way of writing this is a
1:51:03and a or b and this whole thing will
1:51:07become a right. So this one and this one
1:51:11both are basically the law. What the
1:51:13this one is trying to say that this is
1:51:15saying that a or a and b it is logically
1:51:18equivalent to a and then a and a or b
1:51:21and then it is logically equivalent to
1:51:23this one is also logically equivalent to
1:51:25a this one is also logically equivalent
1:51:27to a let's try to find it out how we are
1:51:30getting or how we can try to solve it
1:51:32where a or a and b is logically
1:51:35equivalent to a and this one a and a or
1:51:37b is logically equivalent to a let's try
1:51:39to go and let's try to find it out so
1:51:41for example Example suppose you'll try
1:51:43to understand like this
1:51:46how to identify the absorption law
1:51:48because absorption law is very much
1:51:50important. Whatever is written outside
1:51:55that also has been written inside.
1:51:57Whatever has been written outside has
1:51:59also written inside and your operators
1:52:02are different and your operators are
1:52:04different. For example, if your operator
1:52:08outside it is or then inside you'll have
1:52:10the and condition. So this is the main
1:52:14criteria to understand the absorption
1:52:17law. Whatever is written outside it is
1:52:20also written inside and then operators
1:52:22are different and your operators are
1:52:24different then it will become the
1:52:25absorption law. So let's try to discuss
1:52:27many things related to your absorption
1:52:29law. So for example here I can say that
1:52:32this is a is here and then a is here.
1:52:35Whatever whatever has been written
1:52:37outside it has been written inside and
1:52:39then your operators are different. See
1:52:41it's it it does not depends on b that is
1:52:44why it is called as absorption law
1:52:45because a is absorbing any everything.
1:52:48Whatever has been written outside it is
1:52:50also written inside and then your
1:52:52operators are different. So here a or a
1:52:55and then what you'll get the answer you
1:52:58will be getting is basically what the
1:52:59answer you'll be getting is basically a
1:53:01now suppose what will happen why you are
1:53:04getting this answer for example suppose
1:53:06if I'll put the value of a as a true so
1:53:09the a will become true or a will become
1:53:12here true and now as you can see that
1:53:15the or has seen the true then the whole
1:53:18statement does not matter what you are
1:53:19writing then the whole statement will
1:53:21become what the whole statement will
1:53:23become true. So that means if you'll put
1:53:25the value of a as a true true or true
1:53:27and the whole statement will become
1:53:29true. Now suppose if you put the a as a
1:53:31false here what will get what you'll get
1:53:33false a will become false and then this
1:53:36is or and then this will become false
1:53:39and now as you can see that this and has
1:53:41seen the false.
1:53:44So the left side will become false and
1:53:45here you'll get false. So false are
1:53:48false. Hence the whole statement will
1:53:50become what? Hence the whole statement
1:53:51will become false. So this is the main
1:53:53criteria in order to understand that. So
1:53:56how to you know have the absorption law
1:53:59whatever is written outside it is
1:54:01written inside and then you will get the
1:54:04values of a. Okay that's how you can try
1:54:06to make the concept related to your
1:54:09absorption law. Whatever is written
1:54:11outside it is written inside. Whatever
1:54:13is written outside it is written inside.
1:54:15Now the next one which is called as a D
1:54:17Morgan's law. And then how the D
1:54:19Morgan's law will be looks like
1:54:22sir can eth be seen by the distribution
1:54:25also as well which one you can write as
1:54:288 7th as a you can also try to find out
1:54:31with respect to distribution see suppose
1:54:33if you use the distribution what you'll
1:54:35get a or a and then again and you'll get
1:54:38a or b. So what you can write again the
1:54:41same situations a or a will become a and
1:54:44then a and then this will become a or b
1:54:47again the same situation you'll get this
1:54:49point.
1:54:52If you'll use the distribution you'll
1:54:54get this point. If you use distribution
1:54:55here you'll get this point. So you'll be
1:54:57roaming like this at the end somewhere
1:54:59you have to come up with one value
1:55:00right.
1:55:03Okay let's move to the next one.
1:55:04Absorption law. Uh de Morgan's law. De
1:55:07Morgan's law. This is negation outside
1:55:10and suppose if you'll have this operator
1:55:13when the de Morgan's law enters it
1:55:15changes the operator. So here you'll get
1:55:18negation of A or will become and and
1:55:21then this will become negation of B. And
1:55:24here this is negation A or B. When this
1:55:28negation enters you'll write negation of
1:55:31A
1:55:33or I have written the same thing. For
1:55:36example, here this is negation and then
1:55:38this will become A and B. And suppose if
1:55:41this negation enters here, this will
1:55:44become negation of A or negation of B.
1:55:48Right? So the first one suppose if there
1:55:51is a negation negation A or B and if the
1:55:54negation enters negation of A or will
1:55:56become and and then this will become
1:55:58negation. So do not think that this A is
1:56:01a simple propositional statement. No no
1:56:03no no no these all are compound
1:56:04propositional statement with respect to
1:56:06this also for example suppose if I want
1:56:08to show you the absorption law for
1:56:11example what is your absorption law this
1:56:12is a or and then a and b and then you
1:56:16are getting a how to identify it
1:56:18whatever is outside it is inside and
1:56:20operators are different for example
1:56:22suppose if I will write something like
1:56:23that let's consider a implies b and here
1:56:27a implies b and then or and then this is
1:56:30nothing but c implies d now what is your
1:56:33answer the answer you'll be getting is a
1:56:35imp plus b only basically this is the
1:56:37absorption now right so this you try to
1:56:41understand this whole complex you can
1:56:44immediately you can write as a a implies
1:56:46b so how to identify this identification
1:56:49is very important this box you'll try to
1:56:51consider as a capital a you'll get and
1:56:54this box you'll try to consider this as
1:56:55capital a so again you'll be having this
1:56:57capital a and then you'll get this whole
1:57:00box you'll try to consider this as
1:57:02capital B. So here I'm writing the
1:57:04capital B. So my answer will become
1:57:06what? A understood?
1:57:08So that's how we have to use it. Okay.
1:57:12Now let's go and let's try to
1:57:13understand. See this is related to all
1:57:16the formulas which is related to and and
1:57:18or type of conditions. Now we are moving
1:57:20to the implication. So in implication
1:57:24there is one greatest formula that you
1:57:26people can try to understand. So let's
1:57:28talk about that greatest formula and
1:57:30that greatest formula is for example see
1:57:33whenever you will have a implies b
1:57:38this is also logically equivalent to
1:57:41negation of a or b this is a very
1:57:44greatest you know the formula and trust
1:57:48me this formula is so powerful whenever
1:57:50you will try to
1:57:54you know use this formula in many cases
1:57:57You can solve many questions. Trust me,
1:58:00you can solve many questions. Whenever
1:58:02you will try to use this formula, A
1:58:03implies B, which is logically equivalent
1:58:06to negation of your B. But how this is
1:58:08true? First, you can try to use this
1:58:11with the help of truth table and your
1:58:13truth table. This is your A. This is
1:58:15your B. This is A implies B. And
1:58:17everybody knows that this is true. True.
1:58:19This is true. False. This is false.
1:58:21True. And then this is false. False. So
1:58:24here you'll get true. This is false.
1:58:27This is true. This is true.
1:58:30And then negation of A or B. What is
1:58:33your negation of A? This will become
1:58:35false. False false. True. And true.
1:58:38Right.
1:58:41Lecture
1:58:43lecture lecture.
1:58:46So the first one will become what? So
1:58:49here negation of A or B. So false or
1:58:52true will become true. False or false
1:58:54will become false. true or true will
1:58:57become true and then true or false will
1:58:59become true. So as you can see that this
1:59:01will have the same behavior same column
1:59:04A implies B true false true true true
1:59:06and this is also true false true true so
1:59:09we can say that A implies B is logically
1:59:11equivalent to negation of A or B that's
1:59:13the one point that's the one thing but
1:59:17I'm not trying to focus I'm not trying
1:59:19to tell you key you have to use that you
1:59:22know the truth table every time I can
1:59:25say that this A implies B is logically
1:59:29equivalent to negation of your B. This
1:59:32you can
1:59:34you are using at each and every day.
1:59:38Seriously, every day you're using this A
1:59:41implies B. For example, suppose you are
1:59:44writing the statement key if you go
1:59:49late office
1:59:53then
1:59:55boss may fire you.
1:59:58Boss may fire you,
2:00:01right? Suppose if you're writing A
2:00:03implies B, something like that. If you
2:00:05go late to the office, then boss may
2:00:06fire you. Sometimes we are also writing
2:00:09another way. What we are writing? We are
2:00:12writing that don't go late to the
2:00:15office. Don't go
2:00:19late office
2:00:22or boss fire. Sometimes you're using in
2:00:25our normal language. For example,
2:00:27suppose at your home also you know
2:00:29people used to say your mom used to say
2:00:31my mom used to say that if you wake up
2:00:34late in the morning then I will not give
2:00:36you breakfast. Another way of saying
2:00:38don't wake up in late in the morning or
2:00:41I will not give breakfast. So I gave you
2:00:44explanation with respect to the truth
2:00:45table also and I gave you the
2:00:47explanation with respect to the normal
2:00:48behavior. So this expression is very
2:00:51very important. Do not try to forget. So
2:00:54here you'll have for example this
2:00:59a implies b is logically equivalent to
2:01:02negation of a or b. This is the first
2:01:06type of explanation. And the second type
2:01:09of explanation is a implies b is also
2:01:12logically equivalent to we can write
2:01:13negation of b that will implies negation
2:01:16of a.
2:01:20A implies B is same as writing negation
2:01:23of B that will implies negation of these
2:01:26two are so powerful.
2:01:29These two are so powerful with the help
2:01:31of this we can solve many questions and
2:01:34personally I used to call in my class
2:01:36this as God's rule.
2:01:39God's rule. The second one is a
2:01:40contraositive. I hope you remember that.
2:01:42I personally call this as the God's
2:01:44rule.
2:01:55I personally call this as god
2:01:59where a implies b is same as negation of
2:02:01a or b or another way of writing a
2:02:04implies b is logically equivalent to
2:02:06negation of b that will implies negation
2:02:09of a. These two rules are very much
2:02:12powerful and whenever you don't
2:02:14understand anything whenever whenever if
2:02:17you don't understand anything try to use
2:02:19these two rule 100% it will help you 100
2:02:23and 1% it will help you to understand
2:02:26this
2:02:29anyhow let's move ahead and let's try to
2:02:31solve more questions. So here the next
2:02:33one suppose if I'm writing a implies b
2:02:37and a implies c
2:02:40then this is logically equivalent to it
2:02:44says that a implies which is nothing but
2:02:46b and c.
2:02:49This is your rule a implies b and a
2:02:51implies c. It is logically equivalent to
2:02:54a implies b and c.
2:02:57Now how to solve this or see as I was
2:03:02talking while teaching that this is a
2:03:03type two. So let's consider this is your
2:03:05A and then let's consider this is your
2:03:08B. Let's consider this is your
2:03:09expression A and then this is your
2:03:11expression B.
2:03:14This was the thing I was talking about
2:03:20initially. I told you that left side you
2:03:23will have one expression and right side
2:03:25you'll have one expression. Left side
2:03:26you'll have one type of expression and
2:03:28right side also you'll have one type of
2:03:30expression. You'll have a two different
2:03:31type of expression here. Left side
2:03:34you'll have expression A. Right side
2:03:36you'll have expression B. So we have to
2:03:39check here. This is a logically
2:03:40equivalence. You know the type is there.
2:03:43So we have to use okay we have to use we
2:03:46have to check
2:03:48we have to find it out. Okay, we have to
2:03:51check whether this A and B. Okay, we
2:03:53have to check whether this A and B how
2:03:56they are logically equivalent.
2:04:08I'll try to prove it. So for example,
2:04:10let's consider this a implies B
2:04:15and this one will become A implies C.
2:04:19Now this will become negation of A or B
2:04:22is here
2:04:24and this is negation of A or C. Right?
2:04:28We can use the God's rule. God's rule in
2:04:30in the God's rule the first one will
2:04:32become a negation. So A implies B can be
2:04:35written as negation of A or B and A
2:04:37implies C can be written as negation of
2:04:40A or C. Then what will happen?
2:04:43Now here distributive law you can try to
2:04:49obtain negation of a try to make a
2:04:51string on this negation of a and try to
2:04:54pull it what you will get then you'll be
2:04:57getting which is nothing but b and c
2:04:59here you can write b and c now try to
2:05:01use the godsole in a reverse manner
2:05:05negation of a or b and c what you will
2:05:08get you can say that it will implies a
2:05:10and it will implies b and So it says
2:05:14like that that's how you can understand
2:05:16this clear everyone.
2:05:18So this expression is clear I have
2:05:20solved it. Now let's move to the next
2:05:23one and that everyone please pay
2:05:25attention. So here if I'm writing a that
2:05:28implies c
2:05:31and here I'll write b that implies c and
2:05:36then it is logically equivalent to so
2:05:39for example suppose if I'll write like
2:05:40this a implies c and b implies c. So
2:05:43here I'm writing a implies c and then
2:05:47and it says b implies c. So a imply c
2:05:52what I can write? I can write this as
2:05:54negation of A or C and then and what is
2:05:59this B imply C? B imply C can be written
2:06:01as negation of B or C. Now what we have
2:06:04to do? We have to use or C and then or
2:06:08C. Try to take outside or C. What you
2:06:12will get? You'll get negation of A and
2:06:15negation of B. So if you'll take
2:06:18negation outside what we can write? We
2:06:20can write negation outside and then we
2:06:22can get A or B. Right?
2:06:27A or B or C. Can I write like this?
2:06:32Negation of A or B or C. Now what
2:06:34exactly is a negation of A or B or C?
2:06:36Can I write like A or B or this implies
2:06:39C? Can I say like that?
2:06:46Yes sir. Right. So the answer for this
2:06:50expression you can get it. It says A or
2:06:53B and then this whole thing is implying
2:06:56C. Now one thing you have to make in in
2:07:01in order to understand this is see when
2:07:04the left side is the same here you will
2:07:07have a and here you'll have a when your
2:07:11left side is the same your operator is
2:07:14not changing
2:07:16your operator is not changing
2:07:24but when the right side is the same when
2:07:27the right side is the same then the
2:07:29operator is changing
2:07:32why it is changing this because of D M's
2:07:34law it is changing
2:07:40getting my point
2:07:43yes
2:07:45now if I will write okay I want you to
2:07:48solve this here I have used two and
2:07:51condition but what will happen if I will
2:07:53use or condition so A implies B
2:07:59or A implies C. What should be the
2:08:03answer that you can try to find it out
2:08:05and then again I will write A implies C
2:08:10or then if I suppose if I'll say that B
2:08:12imply C. What should be the answer
2:08:14related to this? You tell me and you try
2:08:17to solve this. So the answer for this
2:08:20one what you're getting left side is
2:08:23same operator will not change.
2:08:29Right side is same operator will change.
2:08:32I hope everybody got the same answer.
2:08:37Right?
2:08:39Yes. Now as you can see that suppose if
2:08:42I'll give you this question here. Okay.
2:08:46Now as you can see that this question is
2:08:49they are not asking the type one they're
2:08:51asking the type two. So if you solve it
2:08:53what you're getting the answer related
2:08:55to this can you tell me
2:08:59see the first thing that you have to do
2:09:01is put the negation inside
2:09:04try to take the negation inside bro I'll
2:09:07solve it see for example here this is
2:09:09your question
2:09:11and it says that negation of P first it
2:09:15is a negation and then it will have P or
2:09:18and then this will become negation of P
2:09:21and here it will become Q and something
2:09:23like that. What you can do? Try to take
2:09:24this negation inside. This will become
2:09:26negation of P. The or will become and
2:09:30and this negation will come here. This
2:09:32is negation of P and Q. Now if this
2:09:36negation will go inside what you can
2:09:38get? You'll get negation of P. This and
2:09:41will be there. But here this negation
2:09:42will go it will become P and then or and
2:09:46here it will become negation of Q.
2:09:48Now as you can see that this is negation
2:09:51of P. This is and and then this will
2:09:54become P or negation of Q. Now take this
2:09:58you know the operators are different. If
2:10:01the operators are different we have to
2:10:02use the distributive law. Now this is
2:10:04negation of P
2:10:08and P
2:10:11or this negation of P
2:10:14and negation of Q. Now negation of P and
2:10:17P. So one of them will become always
2:10:20false. So here you'll get false or
2:10:24negation of P and negation of Q. So
2:10:27false or it is totally depend on this.
2:10:30So negation of P and negation of Q. So
2:10:34oh yes the option D will become right
2:10:37answer. Clear?
2:10:41Yes sir.
2:10:44So type two you have to solve like this.
2:10:47You have to use all the method. You have
2:10:49to use all everything and then based on
2:10:51that you have to find it out. Based on
2:10:53that you have to tell me.
2:10:56I hope that this has been clear to
2:10:58everyone.
2:11:07Try to solve this also
2:11:12and get it right. Done both the
2:11:16questions.
2:11:19Yes or no? Yes sir.
2:11:21Yes sir. What is the answer for the
2:11:23first one?
2:11:25A
2:11:27PNT. PN P and
2:11:32right P and D. Second one. And for this
2:11:36a
2:11:46Did you solve this one?
2:11:49A or B?
2:11:52Explain. Second answer.
2:11:54Second one. Yes sir.
2:11:58Second one or first one?
2:12:02Could you solve fish? Sir, can you solve
2:12:04second one? Okay. Sir, can you solve
2:12:07first one, sir? Okay, I'll solve both.
2:12:10See here, this is P or
2:12:15P and Q,
2:12:19right? One bracket and then here one
2:12:21bracket is also there or
2:12:24this is P and Q and negation of R. First
2:12:30I will solve this bracket. Now as you
2:12:32can see that whatever is outside it is
2:12:35inside. So this whole expression will
2:12:37turns out to be P
2:12:39or and then P and and then Q and
2:12:44negation of R. Again whatever is outside
2:12:47it is inside operators are different. So
2:12:49again this whole equation will become P.
2:12:51The same situation will also be happened
2:12:53here. As you can see that this whole
2:12:56expression will turns out to be a P and
2:12:59now this T is outside. You can also
2:13:02write like this. This is T or
2:13:04commitative.
2:13:06Now this you can write P and R and T.
2:13:10Now whatever is outside it is inside and
2:13:13then operators are different. So this
2:13:15whole equation will turns out to be T.
2:13:17So that is why you'll get P and T. Clear
2:13:20everyone?
2:13:22Yes sir.
2:13:25Okay. Now let me see the second one.
2:13:27Second one. Second one what it says? It
2:13:30says here you will have P. This is and
2:13:34and then negation of R or Q or negation
2:13:39of Q. So first we will try to solve
2:13:41this. Now if you will try to solve this
2:13:43what you are getting? One is positive,
2:13:46one is negative. So whenever there is a
2:13:48one is positive, one is negative you'll
2:13:50always get one is false. Why? As you can
2:13:52see that see this is P or negation of P.
2:13:56Either you can suppose if you put the
2:13:58value of P as P as a true. P can have
2:14:01only two values. Either P can be true or
2:14:03P can be false. Suppose if you're
2:14:05putting putting P as a true then true or
2:14:07false answer you will be getting as
2:14:10false. Sorry.
2:14:13Answer you will be getting this as true.
2:14:15Am I right?
2:14:17Yes sir. Yes sir.
2:14:19Now suppose if you put P as a false then
2:14:22what will happen? Suppose if you put P
2:14:24as a false then the first one will
2:14:26become false second one will become
2:14:28true. So again you'll get answer as a
2:14:30true. So if one is positive one is
2:14:32negative you'll always get a true
2:14:34because one of them will always become a
2:14:36true
2:14:38am I right? Yes sir.
2:14:41So here this whole expression Q or
2:14:45negation of Q. So this will become P and
2:14:48negation of R or true. Now what is this?
2:14:54What is the answer of this? You'll get
2:14:57true.
2:14:59True. True because or loves true. So
2:15:03here
2:15:05P and true what is the answer you'll be
2:15:07getting with respect to P P. So this
2:15:11whole expression
2:15:15this whole expression will turns out to
2:15:17be what? P
2:15:20this this whole expression will turns
2:15:22out to be P. The same situation will
2:15:24also happen here. But here this R is one
2:15:27R is positive and negation of R is
2:15:29negative. But instead of writing
2:15:31together they have written
2:15:33differentially. But the operator is
2:15:35same. So you can write you can use the
2:15:37associative law. You can write or
2:15:41negation of R or true. So this thing
2:15:45will become true or true. Again this
2:15:48thing whole thing will become true and a
2:15:50negation of Q. So if you will use this
2:15:53what you will get? You'll get negation
2:15:54of Q. So the first one will become P
2:15:58while the second one will become
2:15:59negation of Q. So P or negation of Q. So
2:16:02that is why you'll get option B. Clear?
2:16:04Yes sir.
2:16:12Did you get the idea how to solve this?
2:16:17Yes sir.
2:16:19And the same the same formula will also
2:16:22be valid in boolean algebra. The same
2:16:24formula will also be valid into uh the
2:16:27set theory. So for example suppose here
2:16:31I'll try to give you one matrix. This is
2:16:34I personally call in my class. This has
2:16:36the god matrix. What is this god matrix?
2:16:39Whenever you will have the and in
2:16:41boolean algebra you'll have a dot in set
2:16:44theory it will have a intersection. And
2:16:46if there is a or in boolean algebra in
2:16:48dist logic you'll have the or in set
2:16:50three you'll have the union and true in
2:16:54mathematical logic here it will become
2:16:56one false in mathematical logic it will
2:16:59become zero. In set theory it will
2:17:02become universal set and here it will
2:17:04become five. So for example I'll try to
2:17:07show you suppose if I have the
2:17:09absorption law a or and then a and b
2:17:13this is absorption law I can easily
2:17:14write as a suppose if I want to make
2:17:17this theorem into set theory I will
2:17:20write a union and then a intersection b
2:17:24suppose if this is a if suppose this is
2:17:27a this is b a intersection b you'll get
2:17:30this part right so a union a
2:17:32intersection b what you'll get you'll
2:17:34get a Suppose if you'll use this thing
2:17:36into modal algebra a plus a dob and then
2:17:40again answer you'll be getting a. So
2:17:43that's how you people can try to use it.
2:17:45So you just have to learn one particular
2:17:47theorem either you can learn with
2:17:49respect to distal logic or you learn
2:17:52with respect to here this formula is
2:17:55valid for logic here this is for boolean
2:17:58algebra and this is for set theory. So
2:18:01you just have to remember one particular
2:18:03formula. If you know about logic you can
2:18:06solve boolean algebra. If you know about
2:18:08boolean algebra you can solve the set
2:18:09theory. So these all are interrelated
2:18:11with each other. So either if you
2:18:14remember one you can remember all
2:18:15yesterday what we have covered I mean we
2:18:18have covered type one and type two and
2:18:20today we will be going and we'll be
2:18:22starting with the type three and type
2:18:24three is related to the inference rule.
2:18:36inference rule.
2:18:38See basically
2:18:41if you will ask me related to type one
2:18:44or type two I'll say that the more you
2:18:45practice
2:18:47the more you will get that answer.
2:18:49That's it. This is the only difference
2:18:51related to your type one and type two.
2:18:55Again the whole logic is based on the
2:18:58concept of
2:19:01practicing.
2:19:02The more you'll practice the more you'll
2:19:04get that answer and it is very simple
2:19:07actually there is no rocket science into
2:19:09this.
2:19:12Whatever the formula that we have you
2:19:14know discussed yesterday again in that
2:19:17also everything is dependent on that.
2:19:22Okay. So let's start with that today's
2:19:23topic and then that is called as a
2:19:25inference rule. What exactly is
2:19:27inference rule? Let me try to understand
2:19:30or let me try to give you an explanation
2:19:32with respect to this related to your
2:19:34inference rule. See this is very simple
2:19:36concepts related to your inference rule.
2:19:39uh in order to understand the inference
2:19:41rule the one thing that you have to
2:19:43understand is basically whatever the
2:19:46concept you are you know remembering for
2:19:49type one just forget type one because
2:19:53when you will be having the type one or
2:19:55when you will try to connect to your
2:19:57type one then you can you'll you you'll
2:20:00not have uh you will not understand the
2:20:03inference rule that's the main concept
2:20:05so the first thing that you have to
2:20:06understand you just have to you know
2:20:08forget about type one we will use see at
2:20:11the end I will tell you whenever you
2:20:13will see a question and then when to use
2:20:16the type one and when to use the type
2:20:17three see for example when I taught you
2:20:20type one and then you practice four to
2:20:22five questions and then you understood
2:20:24that uh sir type one is very easy we can
2:20:27try to solve all the questions with a
2:20:29type one right but when you will see the
2:20:32type three I mean this is your inference
2:20:34rule this is your type three whenever
2:20:35you will see this type three then what
2:20:37will
2:20:38Once you will understand about the type
2:20:40three then you will be having this
2:20:43concept sir. Type three is more easy as
2:20:47compared to type one. But the type one
2:20:49is more advanced actually not advanced.
2:20:52Type one covers many things but type
2:20:54three is having more limitations
2:20:57that you can only understand once you
2:20:59will understand about the type three. So
2:21:00let's go and let's try to understand
2:21:02related to type three which is nothing
2:21:03but your inference rule. So what exactly
2:21:06is inference rule? First we will first I
2:21:08will try to give you the definition of
2:21:09inference rule and then I will explain
2:21:11you. So initially 15 to 20 minutes you
2:21:14will not have that idea. You will be
2:21:16thinking sir what exactly is inference
2:21:18rule and how you're getting that
2:21:19inference rule
2:21:25and I know that you'll have a large
2:21:27amount of problems related to this
2:21:30what exactly is the inference rule and
2:21:32how we can try to solve the questions
2:21:34related to the inference rule. So I know
2:21:36that you'll be having a problem. I know
2:21:39that you'll be having
2:21:42you know the main things related to
2:21:44this. But anyhow what I'm trying to tell
2:21:46you just give me 15 to 20 minutes.
2:21:49Initially 15 to 20 minutes you will not
2:21:52understand anything. But after 20
2:21:54minutes when you will try to connect all
2:21:56the dots this this this when you will be
2:22:00trying to connect all the dots at the
2:22:02end you will get to know that oh this
2:22:03exactly is in. So let's try to
2:22:06understand. See whenever you will see
2:22:08this sign first of all just forget about
2:22:10type one. Whenever you will see this
2:22:12particular sign I'm trying to first
2:22:14define the inference rule and I know
2:22:16that you may have also studied the
2:22:18inference rule in your college but you
2:22:20are just knowing about the formulas
2:22:21modus modus r is but what exactly what
2:22:24is the history behind it? How you are
2:22:26getting modus is valid
2:22:29that I will try to explain you. So what
2:22:31exactly is the inference rule? Inference
2:22:33rule says that whenever you will see
2:22:35this particular sign, whenever you will
2:22:37have this particular sign, just try to
2:22:39understand this particular sign. Okay?
2:22:42Whenever you will see this particular
2:22:43sign, then what you have to do, you have
2:22:46to see what has been written on the left
2:22:48side and what has been written on the
2:22:49right side. See, whenever you will see
2:22:51this sign, whatever has been written on
2:22:52the left side, that is called as a
2:22:54premises
2:22:55and whatever has been written on the
2:22:57right side, that exactly is called as a
2:22:59conclusion. So there are two things
2:23:01basically. One is called as a
2:23:03premisesis. Another one is called as a
2:23:05conclusion. Whatever has been written on
2:23:07the left side is called as a premises.
2:23:09Whatever is written on the right side is
2:23:10called as a conclusion. These two things
2:23:12you have to understand. These two things
2:23:14you have to understand. Whatever it has
2:23:16been written on the left side is called
2:23:17as a premises. Whatever has been written
2:23:19on the right side is called as a
2:23:21conclusion. Now you may have a doubt
2:23:23what exactly is a premises. Let's try to
2:23:25take the example. For example, if I will
2:23:27write a implies b or c. One example. So
2:23:30this one is called as a premisesis while
2:23:32this one is called as a conclusion. So
2:23:34it's a very simple things related to
2:23:36inference. If I will say that a implies
2:23:38a implies b or c. This is just a random
2:23:41I have written random example I have
2:23:43written. Whenever you will say that a
2:23:44implies b or c then your left side is
2:23:47called as a premisesis while the right
2:23:48side is basically called as a
2:23:50conclusion. Now if I will be writing key
2:23:52a and b and then it will implies then
2:23:55this thing will call as a premisesis
2:23:57while this one is called as a
2:23:58conclusion. So it's all about the left
2:24:00and right side.
2:24:02So what exactly is a premises? Premises
2:24:04is nothing. This is also a propositional
2:24:06statement. This is also propositional
2:24:08set. I just need 20 minutes. After 20
2:24:10minutes, I will connect everything.
2:24:12Whatever has been written on the left
2:24:13side is also the propositional
2:24:15statement. Whatever has been written on
2:24:16the right side is also called as a
2:24:18propositional statement. So everything
2:24:19is a propositional statement basically.
2:24:22So as you can see that this over here,
2:24:24this is called as a propositional
2:24:25statement. And whatever has been written
2:24:28on this side, this is also called as a
2:24:29propositional statement. So everything
2:24:30is a propositional statement. Basically
2:24:32my point is very clear. Premises is also
2:24:35propositional statement. Conclusion is
2:24:37also propositional. Both are
2:24:38propositional statement. So it's all
2:24:40about left and right. Whatever the
2:24:41propositional statement has been written
2:24:43on the left side is called as a
2:24:44premises. Whatever the propositional
2:24:46statement which has been written on the
2:24:48right side is called as a conclusion. So
2:24:49that's it. It's all about the left and
2:24:52right. Whatever you are writing on the
2:24:53left side is called as a premises.
2:24:56Whatever you are writing on the right
2:24:57side is called as a conclusion. See your
2:24:59propositional statement can be simple or
2:25:01it can be compound propositional
2:25:03statement. For example, here the left
2:25:05side is a simple propositional
2:25:07statement. Right side is a compound
2:25:08propositional statement. Left side is a
2:25:11simple propositional statement. Right
2:25:12side is a compound propositional
2:25:14statement. Or it can be mixture of
2:25:15anything. For example, it can also be
2:25:17very simple. Simple. This is also
2:25:19simple. This is also simple. Or for
2:25:21example, let's consider key. This can be
2:25:23compound also. So this left side is also
2:25:25compound. Right side is also compound.
2:25:27So it's not just simple or compound or
2:25:30something like that. It's a very basic
2:25:31concept. Whatever you are writing on the
2:25:33left side is called as a premises. And
2:25:36that exactly is a propositional
2:25:37statement. And whatever you are writing
2:25:39on the right side that is also the
2:25:40propositional statement. Right? What
2:25:43exactly is that? That is simple and
2:25:44compound. You have to understand like
2:25:46that. And it's all about the left and
2:25:48right side. It's everything is all about
2:25:51left and right side. Whatever you are
2:25:53writing on the left is called as a
2:25:54premises. Whatever you are writing on
2:25:56the right is called as a conclusion. So
2:25:58what exactly sir what exactly is the pre
2:26:00inference rule? Inference rule says that
2:26:03consider the premises as true. Okay.
2:26:06Consider
2:26:09the premises as true. Consider the
2:26:12premises as true.
2:26:15And check the conclusion. And check the
2:26:18conclusion.
2:26:20and check the conclusion
2:26:24and check the conclusion. So you have to
2:26:26consider your premises as true. What
2:26:28exactly is a premises? The left side is
2:26:30a premises and you have to consider that
2:26:33as a true. So consider that left side as
2:26:35a true. When you will consider the left
2:26:38side as a true, you have to check the
2:26:39right side. What exactly is the right
2:26:41side? The right side is what? Consider
2:26:43the primaxis as true and check the
2:26:45conclusion. Now conclusion may have a
2:26:47two things. First suppose if your
2:26:50conclusion is also true then that whole
2:26:53expression will become valid. The whole
2:26:55expression will become valid and then
2:26:57that comes under the inference rule. So
2:27:00you you do not have to do anything. You
2:27:02have to consider the premises as true.
2:27:04When you are considering the premises as
2:27:06true when you are taking your left side
2:27:08as a true what you have to do you have
2:27:10to check the right side. What can be the
2:27:13right side? Right side can be true or
2:27:15right side can be false. Suppose if your
2:27:17right side is coming as a true then
2:27:20there is no point of having the false
2:27:24everything is become true everything
2:27:25will become true. So when everything
2:27:27will become true the conclusion will
2:27:28become a true and if the conclusion will
2:27:30become a true then this will comes under
2:27:32what? This will comes under the
2:27:33inference rule and then this is very
2:27:35simple actually then whole thing will
2:27:37become what? The whole thing will become
2:27:39valid and then this will comes under the
2:27:40inference rule. What will happen if your
2:27:42conclusion will become false? So as soon
2:27:44as your conclusion will become false
2:27:47then it is not valid then it does not
2:27:49comes under the inference rule. This is
2:27:50a very simple concept.
2:27:53This is one way of writing premisesis
2:27:55and conclusion. Another way of writing
2:27:57you can also write like this. You here
2:28:00this is your premises vertical way. This
2:28:02is your premises and then therefore you
2:28:04can write which is nothing but your
2:28:05conclusion. So here you can have this
2:28:07conclusion. So you're you can either you
2:28:10can write the horizontal way premises
2:28:12which implies the conclusion or another
2:28:14way of writing here premises and then
2:28:16the conclusion. So either you can write
2:28:17like this or you can write like this.
2:28:19See sometimes you may have a different
2:28:22types of premises. I know that you are
2:28:24thinking sir I'm not getting anything
2:28:26and I told you just give me 20 minutes
2:28:28after 20 minutes everything will become
2:28:29mo everything will become easy actually.
2:28:33Now what you have to consider the left
2:28:35side it can be the multiple premises for
2:28:38example here you will have the premises
2:28:40number one I mean you will have the one
2:28:42type of propositional statement this is
2:28:45you will have the another type of
2:28:46prepositional statement so the left side
2:28:49can have a multiple premises so premises
2:28:51number one premises number two premises
2:28:53number three and let's consider there
2:28:55are n number of premises and then that
2:28:56leads to what and then that leads to
2:28:59which is nothing but what is nothing but
2:29:00one conclusion so here you'll have The
2:29:03multiple premises, premises number one,
2:29:05premises number two, premises number
2:29:07three. There are multiple premises and
2:29:09then multiple premises leads to what?
2:29:11The multiple premises leads to the
2:29:12conclusion. Now what is your main
2:29:15thought about it? Your thought is very
2:29:17simple. You have to consider the left
2:29:20side as a true. Now consider this
2:29:21statement as a true. Whenever you are
2:29:23considering left side as a true and what
2:29:26you have to do, you have to check the
2:29:28right side. Right? So when you are
2:29:30taking your left side as a true okay
2:29:33when you are taking your left side as a
2:29:35true that means I can say that this must
2:29:38also be true this must also be true this
2:29:39must also be true this must also be true
2:29:42and then there will be an and condition
2:29:44between all. So this is one way of
2:29:46writing in a horizontal way or you can
2:29:48also write in vertical way. Let's
2:29:50consider this is nothing but premisesis
2:29:52number one. This is nothing but
2:29:53premisesis number two. This is nothing
2:29:54but primis number three. And then there
2:29:56are multiple primises and that leads to
2:29:58what? that leads to confusion. This is
2:30:00one way of writing and then this is
2:30:01another way of writing. So till now I
2:30:04just told you two things.
2:30:06First one the definition of inference
2:30:08rule. What is the definition of
2:30:10inference rule? It says that take the
2:30:12left side as a true and try to check the
2:30:14right side what exactly it is coming.
2:30:16Right side may be true or right side may
2:30:18be false. So if you're taking your lefty
2:30:21side as a true for example P1, P2, P3 I
2:30:24mean this whole lefty side. Okay. Try to
2:30:27take this left side this whole left side
2:30:28as a true and then check the right side.
2:30:31I mean try to see that what you are
2:30:33getting the right side is true or not.
2:30:38This is one way of writing the left
2:30:40side. Try to consider the left side is
2:30:41it true and try to check the right side
2:30:43is conclusion is coming as a true or
2:30:45not. So this is one way of writing
2:30:49or here this is the definition. Consider
2:30:51the premaxis as true and check the
2:30:52conclusion is coming as a true or not.
2:30:54So you will have a two options. For
2:30:56example, suppose if you're taking your
2:30:57conclusion and if the conclusion is
2:30:59coming as a true then it will become a
2:31:01valid and then we can say that it comes
2:31:03under the inference rule
2:31:08conclusion and then the conclusion it
2:31:10implies true and then true implies valid
2:31:13and then 100% it will comes under the
2:31:15inference rule. 100% it will comes under
2:31:17the inference
2:31:21otherwise it does not comes under the
2:31:22inference rule.
2:31:30Sir I have not understood anything. I
2:31:32know that
2:31:34initially you will not get anything but
2:31:36once I will try to cover many things
2:31:38then you will try to understand
2:31:41many things related to that
2:31:44for example let's consider one
2:31:46premisesis here this is one premises and
2:31:49every time you have to consider your
2:31:51premises as a true for example suppose
2:31:53if I'm writing my mobile phone here this
2:31:55is one premises which I'm see it's not
2:31:57just about the mathematics you can
2:31:59create your own rule into the inference
2:32:01rule
2:32:02Inference rule will gives us powers so
2:32:04that we can generate our own rule. We
2:32:07can make our own rule. But we have to
2:32:10follow the certain things. And what are
2:32:12these certain things? We have to
2:32:13consider the premises as true. Consider
2:32:16the premises as true. So here this is my
2:32:19one premises. I'm taking one premises.
2:32:21I'm trying to generate one rule here.
2:32:23Okay. I'm trying to generate one rule
2:32:25here in a simple way. So let's consider
2:32:27there is one premises and it premises
2:32:29says that mobile phone either it is in
2:32:32left pocket or right pocket okay mobile
2:32:35phone
2:32:37mobile phone it is in
2:32:40left or right pocket left or right
2:32:44pocket okay here this is one type of
2:32:46example my mobile phone it is into the
2:32:50left pocket or right pocket and this
2:32:51statement is true 100% this statement is
2:32:53true there is no doubt into that this
2:32:56the first premises and 100% it is true.
2:32:59It must be into the left pocket or it
2:33:00must be into the right pocket. I mean
2:33:02there is or condition is there. So
2:33:03either it is in left pocket or it is in
2:33:05right. Now here I'm taking the another
2:33:08premises and that premises is also true.
2:33:10It is not in left pocket. It is not in
2:33:14okay it is not in left pocket. It is not
2:33:18in left pocket. For example, let's take
2:33:20this example. It is not in left pocket.
2:33:22And this is also this statement is also
2:33:24true. It is not into left pocket. So
2:33:27what should be your conclusion? Just
2:33:28forget about the mathematics. Just tell
2:33:30me with in a simple manner.
2:33:34Suppose if mobile phone it is in left or
2:33:36right pocket and the second premises
2:33:38which is also true and it says that it
2:33:40is not into the left pocket. What should
2:33:41be your opinion on this? Then 100% must
2:33:44be in right pocket. Then I can say that
2:33:46it must be in right pocket. It must be
2:33:50in
2:33:53the right pocket.
2:33:57Is this true or false? What should be
2:33:59your opinion on this? We can definitely
2:34:01say that the statement will become true.
2:34:04Yes or no? It must be in a right pocket.
2:34:07Here I'm having the first statement.
2:34:09Mobile phone it is in left pocket or it
2:34:11is into the right pocket. This is the
2:34:12first statement and which is true.
2:34:14Second statement it is not into the left
2:34:16pocket. then 100% what should be your
2:34:19conclusion? Conclusion it must be into
2:34:20the right pocket. Now let's try to prove
2:34:23these things with the help of
2:34:24mathematically. So for example here I'm
2:34:26writing my first statement. Okay I I'll
2:34:29try to convert into the logical
2:34:31expression. Logical expression says that
2:34:33mobile phone it is into the left pocket
2:34:35or it is into the right pocket. This is
2:34:37your first expression. Now second
2:34:39statement it says that it it is not into
2:34:42the left pocket. So this is your first
2:34:44primis. This is the second premises and
2:34:46the conclusion you it is showing that it
2:34:48must be into the right pocket. So how we
2:34:50can try to write it down? This we can
2:34:52try to write it down. Here you will have
2:34:54the first premises. This is your second
2:34:55premises and then it leads to what? It
2:34:58leads to the conclusion. So this is the
2:35:01way of writing horizontal way of
2:35:02writing. What is your first premises?
2:35:05Your first premises will become L or R.
2:35:07This is your first premises
2:35:11and this will become the first premises
2:35:14and second premises will become which is
2:35:16nothing but negation of L and what it
2:35:19says it says the conclusion the
2:35:21conclusion is basically R. So now let's
2:35:23try to implement the concept of
2:35:25inference rule here. And the concept of
2:35:28inference rule says that key try to take
2:35:31the left side as a true. Try to take the
2:35:34left side as a true. Okay. Try to take
2:35:36the left side as a true and try to check
2:35:40the conclusion is coming as a true or
2:35:42not.
2:35:45We have to check the conclusion is
2:35:47coming as a true or not. So this thing
2:35:49we have to understand. We have to take
2:35:51the left side as a true. Okay, we have
2:35:53to take the left side as a true and we
2:35:56have to check the right side is true or
2:35:57not. So how we can try to understand?
2:35:59See, let's try to take one truth table
2:36:02here. So I'm having this truth table.
2:36:04This will become L. This will become R
2:36:07and then here you'll get L or R and this
2:36:11is for example let's consider this is
2:36:13negation of L and then here you'll get L
2:36:16or R. Now uh with the help of true table
2:36:19I'll try to show you I'll we have a
2:36:22different method I'll try to show you
2:36:23with the help of this for example this
2:36:26will become true this will become true
2:36:28this will become true this will become
2:36:30false here it will become false this
2:36:32will become true and then this will
2:36:34become false this will become false
2:36:36negation of L what you are getting this
2:36:39you'll get false this you will get false
2:36:42and then this you will get true and then
2:36:44this you'll get true now LR R what it
2:36:47will become true or true here you will
2:36:49get true or false you will get this as
2:36:52true false or true you will get this as
2:36:55true false or false you will get this as
2:36:57false now according to this premises
2:37:00what the premises says that first
2:37:02premises L or R when it will become true
2:37:05it is becoming true here it will become
2:37:08true here it will become true here and
2:37:10where is this negation of L negation of
2:37:12L is becoming true here now according to
2:37:14the inference rule it says I try to take
2:37:17the left hand side as a true. So I am
2:37:19trying to take this as also true and I'm
2:37:21trying to take this as also true and I
2:37:23will try to check the conclusion. What
2:37:25is my conclusion? So if I'm trying to
2:37:27take this this one as a left as a true
2:37:29here it will become true. It will become
2:37:31true. It will become true. When I'll try
2:37:33to take this as true it is it will
2:37:35become true here. True here. But there
2:37:37is a and condition. So I have to take
2:37:39this particular row because this is my
2:37:41first premises. This is my second
2:37:42premises. Now check the conclusion. my
2:37:44conclusion is coming out as a true. So
2:37:47we can say that this will become your
2:37:48inference rule. So if I'm trying to take
2:37:51my left side as a true when I'm trying
2:37:53to take my left side is a true I have
2:37:55checked the conclusion. The conclusion
2:37:57is also coming as a true. So when this
2:38:00happens then it comes under the
2:38:02inference rule. When it comes under the
2:38:04inference rule then the whole expression
2:38:07will become a total logology. It will
2:38:08never become false. It will always
2:38:10become a true.
2:38:12If your R is becoming okay, if your R is
2:38:16becoming false, if your conclusion is
2:38:18coming, okay, I want everyone to use the
2:38:21type one in this and tell me whether
2:38:23this one is toology or not, then it will
2:38:27be helpful. Then you can understand a
2:38:28better way. So any expression any
2:38:31expression which satisfies the inference
2:38:35rule
2:38:37any expression which satisfies the
2:38:39inference rule it comes under uh any
2:38:42expression which satisfies the inference
2:38:44rule then it will become valid. And how
2:38:46it will satisfies the inference rule?
2:38:48Your left side must be true. If you are
2:38:51taking your left side as a true then you
2:38:53have to check the right side and then
2:38:55the right side will also come as a true.
2:38:57When your right side will become as a
2:38:58true then only it will become the
2:39:00inference.
2:39:03Sorry. Yeah, it is a toy. It is a
2:39:05toology from first one. It is a
2:39:08tautology from type one. So that means
2:39:10if you check it is a tology. Why it is a
2:39:13tautology? Because it is satisfying the
2:39:15inference. See it's a very simple here
2:39:16there is a implication when the
2:39:18implication will become false. when the
2:39:20implication will become false. When you
2:39:22will take the left hand side as a true
2:39:24but here if you take left side as it
2:39:26right side immediately is coming out as
2:39:28it. So it will never become a false then
2:39:30it will become a rule.
2:39:39Sorry even if we take the right side as
2:39:41false we can cannot make the left side
2:39:44as true completely. See we are not
2:39:46talking about in inference we never talk
2:39:49about right side we always talks about
2:39:51the left side. So you have to take
2:39:53everything as a left side you have to
2:39:56take inology we need to take from left
2:39:59side true and just we need to take for
2:40:01right side false then we need to
2:40:02compare. Okay. Huh that is type one. Yes
2:40:07sir. Type one see type one is basically
2:40:09to check something. Let me be very
2:40:12clear. Type one is basically to check
2:40:13whether it's a tautology or not. Type
2:40:16three it will gives us power to create
2:40:18our own rule. Here I can create my own
2:40:21rule. I'll show you. So for example what
2:40:23we have done here we have taken L or R.
2:40:28Okay. One way of writing horizontal left
2:40:31or right and then and negation of L and
2:40:36then this whole thing is implying R.
2:40:39This is one way of writing. Another way
2:40:41of writing I can write L or R and then
2:40:44negation of L and then therefore you are
2:40:47getting this as R
2:40:50right or see this L or R do not think
2:40:54that this is left or right this L is one
2:40:57type of compound propositional
2:40:58statement. R is also one type of
2:41:00compound proposition statement. So here
2:41:02as you can see that this is positive
2:41:04this is negative of that positive. So
2:41:07answer you are getting as R. Do not let
2:41:10me be very uh let me give you a point.
2:41:12Do not think that it's like a
2:41:14mathematics you are cancelelling and we
2:41:16are getting answer. No no no no no.
2:41:18Inference rule says that with the help
2:41:21of the first premises and with the help
2:41:23of second premises you are concluding
2:41:27something and that conclusion is R. I'm
2:41:30just trying to help you. One is
2:41:31positive, one is negative. I'm just
2:41:33trying to help you. One is positive and
2:41:35one is negative.
2:41:42So how I can create my own rule? For
2:41:44example, if I will write like this A
2:41:47implies B or C implies D, then negation.
2:41:53If I will write A implies B and then I
2:41:56can write my conclusion which is nothing
2:41:58but C implies D. This is one way of
2:42:01writing. So here I have created my own
2:42:03rule. I can create that particular rule.
2:42:08Because if this is valid then this will
2:42:10also become a valid. Or another way of
2:42:12writing how I can write it here. This a
2:42:15implies b
2:42:19or here I can also write like this
2:42:23a implies b
2:42:26or c implies d.
2:42:30Whole thing will become and and negation
2:42:34of a implies b. This is the first
2:42:37premises. This is the second premisesis.
2:42:39See there is no order. You can consider
2:42:41this is this is also first premises.
2:42:42This is also second premises because
2:42:44there is a and condition in between. So
2:42:46if there do not think like that this is
2:42:48premises one premises true then there is
2:42:50and condition in between. Sorry if it
2:42:52will be the or condition.
2:42:54No there would not be or condition.
2:42:56There will always be and condition
2:42:57between two premises.
2:43:01So, so what this conclusion will say
2:43:03that the your conclusion will become
2:43:05what? Your conclusion will become C
2:43:07implies D. So, this will become your
2:43:09conclusion. It's a very simple concept.
2:43:11This is your conclusion C implies D. In
2:43:14a very simple manner, we can easily try
2:43:16to get it. This is this is this is your
2:43:18C implies D.
2:43:20So, A implies B or C implies D and
2:43:24negation of A implies B. This whole
2:43:26thing is concluding which is nothing but
2:43:28C price and that's how we can make our
2:43:30conclusion
2:43:38getting not getting getting
2:43:43this is the first type of rule
2:43:46we can create your own rule. See I I
2:43:49what I told you if you are writing this
2:43:52for example rectangle and this is your O
2:43:56okay and here you'll have this or
2:43:59condition and suppose if there is a
2:44:01negation and in this suppose if
2:44:05something like this then what should be
2:44:08your conclusion the conclusion will
2:44:09become this O now it's up to you you
2:44:14want to fill 100 variables into this
2:44:17place you can fill it. You want to fill
2:44:20200 variables into this, fill it. For
2:44:22example, if I'll write A implies B or C,
2:44:26B or C. Here if I will write C implies D
2:44:30and X. Now here the negation will become
2:44:34because this is your rectangle. This is
2:44:35also rectangle. So that means whatever
2:44:37you are writing here you have to write
2:44:38here. So this is A implies this is
2:44:40nothing but B or C. What should be your
2:44:42answer? The answer will become C implies
2:44:43D and X. C implies D and X. So this will
2:44:47become your answer.
2:44:51This will become your answer. C implies
2:44:53D and X in a simple manner. So one is
2:44:56positive, one is negative. Do not think,
2:44:59let me warn you, do not think that this
2:45:00is positive or negative.
2:45:03Do not think that this is positive or
2:45:05negative. This is your first premises
2:45:07and this is your second premises and
2:45:10these two premises leads to one
2:45:12conclusion. This is the first premises
2:45:15and this is your second premises and
2:45:17then it leads to the conclusion and then
2:45:20you will have this conclusion like that.
2:45:24See I know that you'll have a doubt.
2:45:26Okay, let me try to explain you just now
2:45:29what we have understood the first rule.
2:45:32If you will say that sir if there is a a
2:45:34or b and if there is a negation of a
2:45:37what should be our conclusion the
2:45:39conclusion will become b. This is one
2:45:41way of writing or you can also write
2:45:43like this. Suppose if you will have a or
2:45:45b and if there is a negation of b then
2:45:49this is also a.
2:45:52So either you can consider like this or
2:45:54you can consider like that. So a or b
2:45:57negation of a answer you will get b or a
2:45:59or b negation of b you will get answer
2:46:01is b. So either you will have like this
2:46:04or you will have like this.
2:46:08Getting not getting m see there is no
2:46:12things to repeat it here. See this is
2:46:14very simple. A or b a or b. For example,
2:46:17what you can write the first expression
2:46:19you can also write like this according
2:46:20to the commitative law. First one you
2:46:22can write B or A and then this one will
2:46:25become negation of B. And what you are
2:46:27getting that answer getting that answer
2:46:28is A. So this is also seen this is I
2:46:32mean your rule you can write like this
2:46:34or you can write like this or you can
2:46:35write like this. Yes. Okay. Let's take
2:46:40this is one way of creating a rule. So
2:46:43you know the one type of rule we have
2:46:44created. Let's try to take the another
2:46:46rule. See here this is my first
2:46:48premises. My first premises says that if
2:46:52perfect matching exist this is the real
2:46:55time example. If perfect matching exist
2:47:00then number of vertices will become even
2:47:03then number of vertices will be even
2:47:06then number of vertices will be even
2:47:10will be even. This is my first
2:47:12premisesis. Now second premises I'll say
2:47:15that there is one graph where the
2:47:17perfect matching exists. Then what
2:47:19should be your conclusion? You tell me
2:47:21what should be your conclusion?
2:47:23What should be your conclusion? You tell
2:47:25me everyone.
2:47:28And then the conclusion will become
2:47:30what? Okay. The conclusion will become
2:47:33what? The conclusion will become the
2:47:35number of vertices will become even.
2:47:39Number of vertices will be will be even.
2:47:43will be even. Now let's try to use the
2:47:46you know the logical expression of that
2:47:48logical expression with respect to the
2:47:50first one. Here you will say that P
2:47:52implies Q. This is the first logical
2:47:54expression where the perfect matching
2:47:56exists the number of vertices will
2:47:58become even. P implies Q. What is the
2:48:00second logical expression? Second
2:48:01logical expression will become P because
2:48:04P is for perfect matching. Because if
2:48:06you're taking this one, okay, if you're
2:48:09considering this one as a P and if
2:48:11you're considering this one as a Q, then
2:48:13what will happen? Here it will become P
2:48:15and here it will become Q. So P implies
2:48:17Q and P. Then what should be your
2:48:19conclusion? Conclusion is Q. This is one
2:48:21way of writing. Another way of writing
2:48:23you can say that like this P implies Q
2:48:26is there and P is there and then the
2:48:29whole thing is implying Q. Now according
2:48:32to the inference rule I will try to
2:48:35consider the left side as a true. I will
2:48:38try to consider the left side as a true.
2:48:40According to the inference rule that
2:48:42means I have to take this statement is
2:48:44also true and I have to take this
2:48:46statement is also true. That means I
2:48:48have to check the conclusion. Now try to
2:48:50use the truth table here. So this will
2:48:52become P. This will become Q and then
2:48:55this will become P implies Q. Now what
2:48:58will happen? This will become true. This
2:49:00will become true. This will become true.
2:49:03Now this will become true. This will
2:49:04become false. And then this will become
2:49:06false. This is false. This is true. And
2:49:09then here you'll get true. This is
2:49:11false. This is false. And then this will
2:49:14become true. Now try to understand this.
2:49:16For example, according to the inference
2:49:18rule, P implies I have to take this
2:49:20statement as a true. That means I have
2:49:22to take this as true. I have to take
2:49:24this as true. I have to take this as
2:49:26true. What should be the second second
2:49:28premises? P. You have to take the second
2:49:30one also true but second one is only
2:49:33true at this point. So what is the
2:49:35combination? I have to see the first
2:49:37row. That means if I will take this
2:49:39premises as a true if I will take this
2:49:41premises as a true the conclusion it is
2:49:44showing every time this is true. I hope
2:49:46that you are taking you are getting that
2:49:48why I am taking only the first row. I'm
2:49:51not taking any other thing but I'm only
2:49:53taking the first row because the first
2:49:56row it's showing me the combination of
2:49:57first premises and second premises true
2:49:59value. I cannot take this because here
2:50:02the second premises will become false.
2:50:03So I will not take this. Here the second
2:50:05premises is coming as a false. So I will
2:50:07not take this concept and here the first
2:50:10premises is showing me false. So I have
2:50:13to take those cases where the first
2:50:15premises and second premises both will
2:50:17become true. When both will become a
2:50:18true then I have to see the conclusion.
2:50:20My conclusion is coming out to be a
2:50:22true. So in that way I can say that this
2:50:26will become the it comes under the
2:50:28inference. Definitely it will comes
2:50:30under the inference.
2:50:36Okay. I I'll show you one beauty of
2:50:39God's root. See this is P plus Q and
2:50:43then this is P. Therefore this is Q. Use
2:50:46that God's rule. what you'll get that
2:50:48god's rule negation of P or Q and then
2:50:50this will P and then therefore you'll
2:50:52get Q can I write like this one is
2:50:55positive one is negative both
2:50:58are focusing on I mean see whatever you
2:51:02are writing here this must be the
2:51:04negation of that but it is already a
2:51:07negation it is already negation of P so
2:51:09that means can I write like this yes
2:51:12understood now because what is our
2:51:15method this is your box and then this is
2:51:17oval. This must be negation of this and
2:51:20then you'll get oval. For example,
2:51:22suppose if I'll write negation of P
2:51:23here, then again it will become negation
2:51:24of negation of P negation of P you'll
2:51:26get P and then you'll get this.
2:51:29Understood? Not understood. Yes, sir.
2:51:31Understood.
2:51:34Everyone?
2:51:38Yes sir. Right. Yes or no? Good. Now you
2:51:42everyone try to understand this point
2:51:45and what exactly it is trying to say
2:51:46that it says that when your left turn
2:51:49left gets matching when left one and
2:51:52left one gets matching then you are
2:51:53getting the answer which is nothing but
2:51:55Q and here you are getting answer which
2:51:57is nothing but what is nothing but Q. So
2:52:00P implies Q and then left side you are
2:52:03having P. So left and left getting
2:52:06matching and then the answer you are
2:52:08getting is basically Q.
2:52:10So you can also try to have your own
2:52:13rule. For example, let's consider this
2:52:15is the box and then there is a
2:52:17implication
2:52:20and here you'll be having this oval
2:52:22thing and now you'll have this
2:52:26box
2:52:29and if you'll take that conclusion
2:52:32the conclusion will become what? The
2:52:34conclusion will become this over. It's a
2:52:36very simple.
2:52:40Now it is a time to show you what kind
2:52:44of example is not comes under the
2:52:46inference rule. So here this is my first
2:52:49premises.
2:52:51This is the premises number one. If I
2:52:54will write key if perfect matching exist
2:53:00then number of vertices will become even
2:53:04then number of vertices will be even
2:53:09number of vertices will be even
2:53:13will be even this is your first primis
2:53:16second premisesis let's consider there
2:53:18is a graph G is having
2:53:22even number of vertices
2:53:24having even number of vertices.
2:53:26Even number of vertices this is nothing
2:53:28but your second primises. The first
2:53:30primises I have written and then this is
2:53:33nothing but what? This is nothing but
2:53:34your second premises. So if perfect
2:53:37matching exists then number of vertices
2:53:39will become even. This is your first
2:53:40premises and second premises it says
2:53:43that G is having even number of
2:53:44vertices. Now can I write my conclusion
2:53:47that G
2:53:50G contains perfect matching? What should
2:53:52be your opinion on this? G contains
2:53:54perfect matching then perfect match it
2:53:57may or may not be. It may or may not be
2:54:00that's a very wonderful answer. That
2:54:02means even if I will consider this
2:54:04statement as a true that if perfect
2:54:06matching exist then number of vertices
2:54:08will become even. If I will consider
2:54:10this first statement as a true and even
2:54:12if I will consider the second statement
2:54:14as a true if I will consider my first
2:54:16premises as a true if I will consider my
2:54:18second premises as a true and if I will
2:54:20check my conclusion the conclusion must
2:54:23be true. But here it is showing
2:54:24sometimes it is true sometimes it is
2:54:26false. So when it is showing sometimes
2:54:28it is true sometimes it is false that
2:54:30means I will not consider this under the
2:54:34inference rule.
2:54:37I will not consider this under the
2:54:39inference.
2:54:49I will not consider this under the
2:54:52information. It's a very simple concept.
2:55:00Let me show you with the help of a truth
2:55:02table what this truth table shows that
2:55:05if perfect matching exist. So I'll try
2:55:08to take this as P implies Q and this
2:55:11will become your first premises. This
2:55:13will become your second premises. Second
2:55:15premises is Q. Now the conclusion they
2:55:18are asking for P. So this is one way of
2:55:21writing. Another way of writing. This is
2:55:24one way of writing. Okay. And another
2:55:26way of writing is basically P implies Q
2:55:31and here you will have Q.
2:55:34Then that whole conclusion
2:55:38it says related to P. Now let's try to
2:55:42have the truth table.
2:55:45The truth table says that this is P.
2:55:47This is Q. This is P implies Q. Here
2:55:52let's consider this is true. True. This
2:55:53is true. This is true. This is false.
2:55:56This is false. This is true. This is
2:55:58false. This is false. True implies true
2:56:01will become true. True implies false
2:56:03will become false. False implies true
2:56:05will become true. This is true. Where is
2:56:07your premises? First premises P implies
2:56:10Q. And where it is getting true? It is
2:56:12getting true here. It is getting true
2:56:14here. It is getting true here. Where is
2:56:16your second premises? Q. Q is getting
2:56:19true here. And Q is getting true here.
2:56:21That means if I will take a combination
2:56:23now focus everyone what it says if
2:56:27perfect matching exists the number of
2:56:28vertices will become even. If I will
2:56:30consider this as true. If I will
2:56:33consider this as true, my conclusion is
2:56:35P. That means sometimes it is showing
2:56:38maybe, sometimes it is showing may not.
2:56:41And that's the beauty of this truth
2:56:43table. It shows may and it shows that
2:56:46may not. Okay. Sometimes it may be true,
2:56:49sometimes it may not be true.
2:56:52That is why this does not comes under
2:56:54the inference. Try to use. Okay. Now as
2:56:57we can see that there are different
2:56:59types of rules we have seen. First type
2:57:01of rule is P implies Q and then left
2:57:05side left side P and then therefore okay
2:57:08therefore you are getting answer as a Q.
2:57:11This is one type of rule. Second type of
2:57:14rule P implies Q and if there is a
2:57:17negation of Q and then therefore
2:57:19negation of P. Third type of rule here
2:57:23if I will say that P implies Q and then
2:57:26Q implies R.
2:57:29So that you'll get a conclusion. The
2:57:31conclusion it says that P implies R.
2:57:34Okay. Then it says that P implies R. So
2:57:36P implies Q, Q implies R. And then the
2:57:39conclusion we are getting as P implies
2:57:41R. Now and then the rule we have seen
2:57:45that this is P or Q and negation of P
2:57:49and then therefore you will get Q.
2:57:52Another type of rule suppose if there is
2:57:53a P and then this will become P or Q and
2:57:57if there is a P and Q and then therefore
2:58:00this is P. another rule P or Q negation
2:58:04of
2:58:06Q or R and then therefore you'll be
2:58:10getting which is nothing but P or R. So
2:58:13these all are the different types of
2:58:15rules we are having.
2:58:20Now let's try to understand one by one
2:58:23each one here this is one way of writing
2:58:26horizontal way vert vertical way
2:58:29horizontal way you can also write it
2:58:30like this and what it says it says here
2:58:35this is nothing but P implies Q and P
2:58:40and therefore the whole thing is
2:58:42implying mod Q and second one P implies
2:58:46Q
2:58:48and negation of Q and then this whole
2:58:51thing is implying negation of P.
2:58:54This is P implies Q and Q implies R
2:59:00and then the whole thing is implying P
2:59:02implies R. See, let's try to understand.
2:59:05I don't think you need an explanation
2:59:07with respect to the first one. Let's try
2:59:10to understand the second one. Second
2:59:12one, I'll try to explain like this. P
2:59:14implies Q. What is the contraosity of P
2:59:17implies Q?
2:59:19Negation of Q that implies negation of
2:59:22P. This is a contraositity of the first
2:59:24one. P implies Q. This is nothing but
2:59:27your P. P implies Q. What is the
2:59:30contraositity of this? Contraositive is
2:59:32negation of Q will implies P. And what
2:59:35is the second premises? Second
2:59:37premisesis will become negation of Q and
2:59:39therefore you will get negation of P. So
2:59:42basically that is the same thing.
2:59:46And third one is a transitive rule. I
2:59:48don't think you need any explanation in
2:59:50order to understand the third rule. P
2:59:52implies Q. Q implies R and that's how
2:59:54you're getting P implies R.
2:59:57Let's talk about this one. This I have
3:00:00already explained you and I don't think
3:00:02you need any explanation related to this
3:00:04P or Q and negation of P. Therefore the
3:00:09whole thing is implying Q. Second one
3:00:12here P implies P or Q. Now see what the
3:00:17inference rule says. See this one is
3:00:19important. Try to understand this.
3:00:22What the inference rule says? Inference
3:00:24rule says that take the left side is a
3:00:27true and try to check the right side is
3:00:28coming as a true or not. Now if I will
3:00:31take this as true then what will happen?
3:00:35Then P will become true. When the P will
3:00:38become true and true will see or what
3:00:42will happen?
3:00:44What will happen?
3:00:47True. Everything will become true. Yes
3:00:49or no? Yes sir.
3:00:52Yes sir.
3:00:55So this is one type of truth. Now
3:00:59so I can write like this.
3:01:02Here it says that P implies P or Q.
3:01:07Now here this I can say that P and Q and
3:01:12then this whole thing is implying P. Now
3:01:14try to take the left side as a true.
3:01:16When you will take the left side is a
3:01:18true P and Q. That means if you will
3:01:21take left side as a Q P and Q what you
3:01:23are getting? You are getting both as a
3:01:25true. Left side will only become a true
3:01:27when both will become a true. That means
3:01:29at the same time P is also true and at
3:01:31the same time Q is also true. When P is
3:01:34also true then what will happen? This
3:01:36will become true. Definitely this will
3:01:38become true
3:01:41or we can also use Q instead of P,
3:01:43right? Ah yes, you can use the Q instead
3:01:45of P. That's also good.
3:01:53And here one is positive, one is
3:01:54negative, you'll get P or R. So we are
3:01:57having the different name to this. If
3:02:00you want you can use the name. First one
3:02:02is modus.
3:02:04Here
3:02:10mod stolance
3:02:18here hypothetical serismalism
3:02:31Here
3:02:38addition,
3:02:44this is simplification
3:02:52and this last one is resolution.
3:02:59So when you will when you are
3:03:01comfortable into this I mean if you know
3:03:04all these rules you can easily try to
3:03:07solve any type of questions which is
3:03:09related to your inference rule and this
3:03:11exactly is inference rule. It is nothing
3:03:13you just have to consider the left side
3:03:15as a true and check the right side is
3:03:17coming as a true or not. So once you
3:03:20will solve this you can easily get it
3:03:22everything all the answers related to
3:03:25this. You just have to remember these
3:03:27two things. I mean if you remember this
3:03:29slide very clearly you can understand
3:03:33you can solve any questions and trust me
3:03:37whatever the questions which is based on
3:03:38the basics of logic
3:03:42everyone can easily try to solve that
3:03:45everyone can easily try to solve that
3:03:49let's try to solve some question so for
3:03:51example suppose if you'll have some
3:03:53conditions like this P P implies Q and
3:03:56negation of Q or R and let's consider
3:03:59suppose if they are saying that try to
3:04:00check whether this is valid or not see
3:04:03sometimes they will ask in the question
3:04:06in a horizontal way or they can also ask
3:04:08the questions in vertical way P and P
3:04:11implies Q and the negation of Q or R and
3:04:15then this whole thing is implying R so
3:04:17let's try to use the first one first one
3:04:19says that first premises is P which is
3:04:22given second premises is P implies Q now
3:04:25left side left side is matching you will
3:04:28get Q and then how you're getting you're
3:04:30getting as a mod response. Now the third
3:04:33one which is given you will get negation
3:04:35of Q or R. So one is positive one is
3:04:38negative you will get R and then this
3:04:41one is given. So again you have to use
3:04:43the
3:04:44either you can use disjunctive or modus
3:04:48no problem you'll get R easy.
3:04:52Uh first solve question number eight and
3:04:55tell me it is valid or not.
3:05:00Question number eight.
3:05:03Valid sir.
3:05:05Valid sir or sir. Yes sir.
3:05:09Valid. Okay. Solve question number six.
3:05:14Six. Six.
3:05:19Right. Okay. See always how to solve
3:05:23this type of questions. You have to see
3:05:25the questions and you have to based on
3:05:27uh uh based on the conclusion you have
3:05:30to behave. See for example suppose if
3:05:33this question will come then how I will
3:05:34solve I have to see the conclusion. See
3:05:38uh whenever you want to solve any type
3:05:39of questions always try to see the
3:05:41conclusion. What is the conclusion? That
3:05:43the conclusion is basically W. But where
3:05:45is W? W it has been given on the uh
3:05:48third premises. This is the first,
3:05:50second, third. See it is not necessary.
3:05:52This is the third premises. You can
3:05:54interchange. This can also be the first.
3:05:55This can also be the second because
3:05:57there is and condition in between all.
3:05:59So if there is and condition in between
3:06:00all, you can use at any point of time.
3:06:03My point is very simple. My conclusion
3:06:05is W. So where it has been given? It has
3:06:07been given here. That means T implies W.
3:06:10So if I want to make W as a free, I need
3:06:15T. So if I can get a T then T and T left
3:06:19side, left side matches and then I will
3:06:21get the right side W will be free. But
3:06:23where is T? T is given here but it is
3:06:26not given in the form of T but it has
3:06:29been given in the form of negation of T.
3:06:30I can use a contraositive. Contraositive
3:06:33I can get R implies tree. That means
3:06:37where is T? I need T. That means if you
3:06:40need T then T has been given in the left
3:06:42side or right side. T has been given in
3:06:44the right side. That means you need R.
3:06:46But where is R? R has been given here. R
3:06:49or S. So you if if you can make R as a
3:06:52free then you can apply R here you can
3:06:55get T. You can apply T here you can get
3:06:58W. So let's try to find it out. So here
3:07:01first I will take negation of S and try
3:07:04to use the second one R or S. One is
3:07:06positive one is negative. You'll get R.
3:07:09Try to use a contraositive of this. R
3:07:11implies T. Left to left matches you will
3:07:14get right T and then you'll try to use
3:07:16T. T implies W. Left to left matches
3:07:19you'll get implication is a W.
3:07:21Understood? Now you don't need to
3:07:22remember any questions. You don't
3:07:24remember you don't need to remember any
3:07:26type of answers. I mean you don't need
3:07:29to remember any any type of formula or
3:07:32something like that.
3:07:37Try to solve question number nine.
3:07:42P2 Q2R what it will signify P2R
3:07:47right but here is negation of R that
3:07:48means you can take contraositive
3:07:50negation of R will implies negation of P
3:07:53negation of R you will take and then
3:07:55answer you'll be getting is negation of
3:07:56P we can say that it is also valid
3:08:05question number
3:08:12explain this. So it is to it is totally
3:08:15based on a practice.
3:08:19The more you'll do the practice, the
3:08:20more you'll get that answer.
3:08:25Solve question number four.
3:08:31Okay. I I think I think you need to give
3:08:33more explanation on this. Let me tell
3:08:35you one important point here. You can
3:08:37use P two times. For example, if okay P
3:08:42and P if you will use you will get R
3:08:44implies S. And here you will get what is
3:08:48the contraosity of the first one?
3:08:49Controposity of the first one you will
3:08:51get P implies R. P implies R. R implies
3:08:54S. These two you can write P implies S.
3:08:56You can use P one more time and then
3:08:58therefore you will get S. Do not think
3:09:00that P will vanished. No will P will not
3:09:03be vanished. Sir, we can use any number
3:09:06of times in the you can use any number
3:09:09of times. And then the reason behind it
3:09:11is basically something like that. For
3:09:13example, suppose if I will write like
3:09:16this A and A what should be your answer?
3:09:19You'll get A. A. A and A and A. What
3:09:23should be your answer? You will get A.
3:09:24For example, if I will write A and B.
3:09:26Either I can write a or a and a or I can
3:09:30write a and a or I can write a and
3:09:32because already know that there is a and
3:09:34condition in between. So even if you
3:09:36will use these two condition what you
3:09:38will get you will get b. So you can use
3:09:40a as many s times a. So not a problem
3:09:42because there is and condition. So a and
3:09:44a is a a and a and a is a. So that is
3:09:47why here if even if it is one times p
3:09:49you can use it two times three times. No
3:09:51not a problem. Do not think that this p
3:09:53will vanish. Let me tell you a very
3:09:55simple thing. This is not a mathematics
3:09:57key 5 + 2 will become 7. So five is
3:10:00vanished. No, it's like there are three
3:10:04things and these three things are
3:10:06implying something and that implication
3:10:10is s.
3:10:13Logical equivalence is different. In
3:10:15logical equivalence you can do like
3:10:16that. Logical equivalence is something
3:10:18different. Here logical equivalence says
3:10:21that left side is equals to right side.
3:10:23Here the left side is not equals to the
3:10:24right side. your left side is implying.
3:10:26So I need to give a clarification on
3:10:29this because many times the students are
3:10:31having a problem. For example, if I will
3:10:33write key a implies b, it is logically
3:10:36equivalent to negation of a or b. So
3:10:38this is your left side and left side is
3:10:41equals to right side. Here if I will
3:10:43write a and a implies b and then this
3:10:46whole thing is implying b. So what it
3:10:49says? It says that this primis this
3:10:51premises this left side is implying
3:10:54something and that something is called
3:10:56as a b. So implication is different
3:10:59thing while equivalence is a different
3:11:02thing. Equivalence means whenever you
3:11:04have a implies b in place of a implies b
3:11:07you can write negation of a or so you
3:11:10can you can delete it. For example
3:11:13suppose if you will have x it is equals
3:11:15to 5 and if I will write 2x + 3. So in
3:11:18place of x I can write five. So 2 5 and
3:11:21then plus three. This is mathematics
3:11:23because x is equals to 5. So in place of
3:11:25x you can write five. That is valid.
3:11:28But here this saying that this whole
3:11:32thing is implying b. So it is implying
3:11:35it is not equivalent.
3:11:37Implying is a different thing.
3:11:38Equivalence is a different thing. There
3:11:40is a difference into that. Clear? Today
3:11:43I will tell you where to use the type
3:11:45one and where to use the type three. So
3:11:48by seeing the questions you can easily
3:11:50try to get it. Okay.
3:11:55Now if you will try to see the first
3:11:58question in the first question there is
3:12:01no need to use the inference rule and
3:12:04there is no need to use the
3:12:07this concept. uh what we can say we
3:12:11don't need to use the concept of type
3:12:13one we don't need to use the concept of
3:12:15type three in the first one you know why
3:12:25because this question is related to as
3:12:28you can see that this question the first
3:12:30question is related to this uh if the
3:12:33right side is same
3:12:36then the operator will change. This
3:12:39question is belongs to logical
3:12:40equivalence. Am I right?
3:12:44Yes.
3:12:46Right. Okay. Let me tell you a very
3:12:48beautiful thing. Whenever there is a a
3:12:51double implication b, so a double
3:12:53implication b, it is logically
3:12:55equivalent to that means a can imply b
3:12:59and at the same time b can also imply a.
3:13:03So a can imply b and at the same time b
3:13:07can also imply a. This is the beauty of
3:13:10double implication. This is the beauty
3:13:12of logical equivalence. Whenever there
3:13:14is a a a double implication b that means
3:13:17a can imply b and b can imply a. Now we
3:13:20know that this part
3:13:23this part is logically equivalent to
3:13:26this particular part. So that means the
3:13:29a can imply b and b can also imply a. So
3:13:33both can A can also imply B and B can
3:13:36also imply A.
3:13:41Understood the first point?
3:13:45Yes sir.
3:13:47Okay. What about second one? Second one
3:13:49is valid or not? You can use the WS
3:13:52rule. Negation of P or Q,
3:13:56negation of R of S and then P or R. If
3:14:01you'll combine these two resolution, one
3:14:04is positive, one is negative. What
3:14:06you'll get? You'll get P or S. Q. Next
3:14:10one P or Q. One is positive, one is
3:14:12negative. Again, that means you'll get Q
3:14:16or S. So, the second one is valid. Easy.
3:14:22Yes or no? Yes, sir. Okay. What about
3:14:26this?
3:14:31Can you repeat the first one please?
3:14:34First one. This one. Yes sir.
3:14:38This one is basically if you remember
3:14:40that uh in the type two logical
3:14:42equivalence when I was teaching you the
3:14:45concept there I taught you this box is
3:14:48equivalent to this box. Whenever one box
3:14:51is equivalent to another box that means
3:14:54equivalent means double implication. So
3:14:56whenever there is a double implication
3:14:58that means if A is logically equivalent
3:15:02to B or if A is double implication to B
3:15:06that means A can also imply to B and B
3:15:08can also imply to A. So we know that
3:15:13this is our box A and box B. So A and A
3:15:17and B both are same. So A can imply B
3:15:20and B can imply A. So here A is implying
3:15:22B.
3:15:25So if it is in the box we need to
3:15:27consider it is in double.
3:15:30Yes.
3:15:32Okay.
3:15:38Okay. Fine. Any problem with respect to
3:15:41this slide? Please let me know. This is
3:15:42a gate question and uh we have already
3:15:46solved this gate question with a type
3:15:48one
3:15:50but I want you to use the type three.
3:15:52Now see the first question.
3:15:56See whenever you will whenever there is
3:15:58a implication try to use the god's rule.
3:16:01The god will help you. That is a very
3:16:03simple concept.
3:16:05So what you can use you can write
3:16:07negation of p or q. Second r is implying
3:16:11s that means negation of r or s and that
3:16:15means here you will get p or r.
3:16:18Yes sir. So one is this one one is
3:16:21positive one is negative you'll get P or
3:16:23S here this is negation of P or Q one is
3:16:27positive one is negative what you will
3:16:30get you'll get Q or S now if you use the
3:16:34God's rule here you will get S or Q so
3:16:36that is why the first one is valid
3:16:40right
3:16:44yes sir
3:16:46okay
3:16:48what What about S?
3:16:50See the option S. By seeing the option
3:16:53S, can we solve this? Yes sir. I don't
3:16:56want to use any
3:16:58R R P
3:17:01for example P implies R P left to left
3:17:05match you'll get R so R will be free
3:17:08here right and then this is P and
3:17:11therefore you'll get R. If you'll apply
3:17:14R here one is positive one is negative
3:17:16you'll get Q. So that is why we can
3:17:18write this as valid. So within 3 second
3:17:20you can solve the question.
3:17:23If you have the if you have the practice
3:17:26of inference rule within 3 second you
3:17:28can solve the question. So that is why
3:17:30the first one this one is valid. This
3:17:32one is also valid. What about the Q?
3:17:37What about Q
3:17:39R as an conclusion? Yes. But what they
3:17:43are expecting for example the first
3:17:46premises as you can see that what they
3:17:48have written this is they have written
3:17:49is negation of P and Q. Now you tell me
3:17:53can I write negation of P and Q as
3:17:56different different I can write negation
3:17:58of P here and I can write Q here. So
3:18:00when we are writing vertically can we
3:18:02say that there is a hand condition in
3:18:04between? Yes sir. Right. So I can also
3:18:08write individual. Now here we will have
3:18:10this Q and then that will implies which
3:18:12is nothing but P implies R. Now use the
3:18:15Q here. So if you'll apply mod exponents
3:18:18what you'll get? You will get P implies
3:18:20R. But here this is not P. This is
3:18:22negation of P. So try to use a
3:18:24contraositive negation of R that will
3:18:26implies negation of P. If I'm using
3:18:29negation of P here then what will
3:18:31happen? Negation of R. Can we write like
3:18:33this? No. This is invalid. Why it is
3:18:36invalid? Because I told you matching the
3:18:39left side and left side you will get the
3:18:41right side but matching the right side
3:18:44you will not get the left side.
3:18:46Understood
3:18:48sir in Q we can take that one as
3:18:50premises and Q one as a conclusion in
3:18:53negation of P and Q. So negation of P or
3:18:58Q. Huh? You can take this as one primis.
3:19:01You can because there is already and
3:19:03condition in between
3:19:05within the brackets also. We need to
3:19:06consider the different things. H
3:19:10or you can use a simplification rule.
3:19:12Negation of P and Q it can imply
3:19:15negation of P one time. Negation of P
3:19:18and Q it can also implies Q. So which is
3:19:20also same thing.
3:19:36clear
3:19:39sir how negation are implies negation P
3:19:43cancel
3:19:45it I'm not cancelelling I'm just trying
3:19:47to say that negation of P and Q you can
3:19:50take two different premises see for
3:19:52example Okay, try to understand in that
3:19:55manner. So for example, if you will have
3:19:58A, if you'll have B and then for example
3:20:00A is implying C. So either you can write
3:20:04like this. This is one way of writing or
3:20:06you can also write like A and B and then
3:20:09here A is implying C. So this way of
3:20:13writing is also same. This way of
3:20:15writing is also same. So this way of
3:20:17writing this way of writing is also
3:20:19same. This way of writing is also same.
3:20:21Both are basically same.
3:20:27There is no difference into that because
3:20:29when I was teaching you then what you
3:20:32have see for example this is the
3:20:34vertical way. If you write horizontal
3:20:35way what you will get? You will get a
3:20:37and b and and then a implies. So
3:20:41automatically this will become a and b
3:20:44because there is an and condition
3:20:45between every premises right.
3:20:58Got it. Yes, sir.
3:21:16So here we can use that and try to use
3:21:19this rule R you will not get the same
3:21:22problem that
3:21:25yes the same problem you are getting the
3:21:28question number five the same here so
3:21:31this is also invalid
3:21:33you will not get R you'll get more than
3:21:36that
3:21:38So that is why the real answer you'll
3:21:40get this one and then this one clear
3:21:42everyone here you'll get okay now you
3:21:45try to see the first question and second
3:21:48question everyone try to see the first
3:21:49question and second question now in the
3:21:52first question just scan it and tell me
3:21:55whether you will use the type one or you
3:21:58will use the type three
3:22:02scan the first question for example in
3:22:04place of you if I will be there so what
3:22:07they have written they have written P
3:22:08implies Q and here they have written P
3:22:11so therefore I will write Q and then you
3:22:14will get R that means at the end I'm
3:22:16getting Q and R but this one is
3:22:18something different right so immediately
3:22:21I'll switch to type one so first I'll do
3:22:23the scanning and I will get to know that
3:22:26okay I will I cannot use the inference
3:22:27rule then I immediately I'll switch to
3:22:30type one so now you can use the type one
3:22:32in the first one did you understand when
3:22:34to use the type one when to use type one
3:22:36or type
3:22:38Clear? Yes sir. Cheers sir.
3:22:41Second question just do the scanning and
3:22:43tell me whether you'll be using the type
3:22:45one or what you'll be using. For
3:22:47example, see I'll tell you a very simple
3:22:50concept in inference rule. You just scan
3:22:54it and try to see that the adjustment
3:22:58you're getting it or not.
3:23:01So immediately if you're getting that
3:23:03adjustment then then and going for
3:23:05example suppose
3:23:08if you if you're require if you require
3:23:11more than 10 second to solve the
3:23:14questions based on inference rule then
3:23:16you have to use the type one for example
3:23:18first I'll see that okay this is P
3:23:20implies Q and P so if I will match P and
3:23:23P okay I'll write like this P implies Q
3:23:26and P then therefore I will get Q and
3:23:28then there is and R so I'm getting Q and
3:23:31R but here they are expecting something
3:23:33different I'll switch to type one then I
3:23:35will use a type one now let's see the
3:23:37second question here okay second
3:23:39question uh something they have written
3:23:43for example let's go to option C so here
3:23:46I have written P or Q or R and then this
3:23:48is negation of Q so one is positive one
3:23:51is negative I will get P or R within 10
3:23:53second I will get so I'll write this is
3:23:55true I will use the in now let's see the
3:23:58second one here so in the second one I
3:24:00am not able to adjust
3:24:04uh but if I will try I will get it. So
3:24:06instead of spending time with respect to
3:24:08type three I will use the type one.
3:24:11Understood? Yes sir.
3:24:16Clear everyone. Yes sir. Good.
3:24:22Try to solve this what you are getting.
3:24:26Yes. You have to use the inference rule.
3:24:28For example, negation of P or Q and R
3:24:31and then R implies S. Transitivity if
3:24:34you use you will get P or Q that will
3:24:36implies S or T.
3:24:40From simplification take negation of U.
3:24:43Negation of U. From simplification
3:24:44negation of U that will implies negation
3:24:46of T. Negation of U, negation of U will
3:24:48get negation of T.
3:24:51and uh right you'll get negation of P
3:24:55and negation of S also I can borrow so I
3:24:58can get negation of S and I can get
3:25:01negation of P use the
3:25:06contraositive
3:25:08negation of S that will negation
3:25:11negation P2 Q sir
3:25:15huh it's getting notation of P2 Q
3:25:20P2
3:25:23You will get P and Q P and negation of
3:25:26Q. You'll get this. Did you understand
3:25:29till now? Ah, yes. Not understood. Yeah.
3:25:32Yeah. Understood. Yes, sir. It's clear.
3:25:35Understood. Please explain. P. So it
3:25:39implies P and negation of Q. That means
3:25:42if you will take left side negation of S
3:25:45and negation of T here, left me match
3:25:47you'll get P and negation of Q. And from
3:25:50simplification you can take P outside.
3:25:52So it will become valid. It is valid. If
3:25:56you're not getting anything please tell
3:25:58me I will I'm here to help you. Don't
3:26:00worry about it.
3:26:03The question the premise is what they
3:26:05have written is negation of
3:26:11is negation of P or Q and then that will
3:26:14implies R and that implies R. Second R
3:26:19that will implies S or T. If you'll
3:26:22combine these two A to B b B B B B B B B
3:26:24B B B B B B B B B B B B B B B B B B B B
3:26:24B B B B B B B B B B B B to C what you
3:26:25will get? You will get negation of P or
3:26:27Q and then that will implies S or T. If
3:26:31you'll take a contraosity of this,
3:26:33you'll get this negation S or T that
3:26:37will implies negation negation of P or
3:26:40Q. This will become negation of S and
3:26:43negation of P. That will implies P and
3:26:46negation of Q. Till here any problem?
3:26:51No problem. Right? No problem.
3:26:55Okay. Now here we are having negation of
3:26:59S. Can I write these two as a two
3:27:01different thing? Yes sir.
3:27:05Now here this will become negation of U.
3:27:07That will implies negation of T. Now if
3:27:10you will combine these two negation of u
3:27:13negation of u what you will get you will
3:27:15get negation of t that means at one
3:27:17place you are having negation of s at
3:27:20the second place you will have negation
3:27:21of t can I say that there is and
3:27:23condition in between yes sir so that
3:27:27means if I will take here then negation
3:27:29of s and negation of t here we are
3:27:32already getting so modus I will get p
3:27:34and negation of q simplification I can
3:27:37write p
3:27:40Oh
3:27:43now it is clear to everyone
3:27:46the man shared one file and in that
3:27:52what the question is like this
3:27:55f_sub_1 and f_sub_2 it implies
3:28:01f_sub_1 and f_sub_2 it implies f_sub_3
3:28:06f_sub_1 and f3 it implies f_sub_3. Okay,
3:28:09f_sub_1 and f_sub_2 it implies f_sub_3
3:28:13and f_sub_1 and f_sub_2 it also implies
3:28:17negation of f3. So the question is
3:28:20saying that this is also tology and this
3:28:23is also tology. That means it cannot be
3:28:26false at any point of time and this one
3:28:29will also cannot be false at any point
3:28:31of time. That means what will happen?
3:28:34That means we have to take the cases
3:28:36basically in this they're asking.
3:28:39So for example
3:28:41suppose if I will put the values of
3:28:44f_sub_3 as a true right case one
3:28:50case one if I will put the values of
3:28:52f_sub_3 as a two. So that means f_sub_1
3:28:55and f_sub_2 it implies true here
3:28:59and f_sub_1 and f_sub_2 it implies false
3:29:03here. But we know that everything is
3:29:06true. That means that means this is also
3:29:09true. And then that means this is also
3:29:11true. Now my point is if this right side
3:29:15is false then 100% this must be a false.
3:29:18False. If the right side is true then
3:29:21because if you're taking this as false
3:29:23you have to take this as false. That
3:29:26means if this is false it will have
3:29:27three cases to get false. This true
3:29:30false false true and false and false.
3:29:34That means
3:29:36if for example here even if I will take
3:29:41all the cases of this there would not be
3:29:43any problem still it will become
3:29:46because it does not matter on the values
3:29:49of for this it does not matter on the
3:29:50values of f_sub_1 and f_sub_2 even if
3:29:53your right side is true it does not
3:29:55matter on f_sub_1 and f_sub_2 so this is
3:29:57your case number one case number two for
3:30:00example you'll take f_sub_3 as a false
3:30:02so the First one f_sub_1 and f_sub_2 it
3:30:07implies false here and f_sub_1 and
3:30:10f_sub_2 it implies true. So basically
3:30:13turns out to be the same condition. I
3:30:15mean case one and case two both will
3:30:16become only one case. He for example
3:30:21f_sub_1 and f_sub_2 the whole summary
3:30:24sometimes it is true and f_sub_1 and
3:30:27f_sub_2 sometimes it is false. So even
3:30:30if it is implying to true, even if it is
3:30:32implying to false. See with respect to
3:30:34this case it does not matters on the
3:30:36values of f_sub_1 and f_sub_2 it will
3:30:38always be true. It will always be true.
3:30:40So hence we have to focus on this
3:30:42particular case. F_sub_1 and f_sub_2 it
3:30:44is false. That means both cannot be true
3:30:46at any point of time. Rest it can have
3:30:49all the cases. So if you will see what
3:30:53you can write
3:30:55f_sub_1 both f_sub_1 and f_sub_2 are
3:30:57tology. No, it is not necessary.
3:31:00And uh f_sub_1 and f_sub_2 is not
3:31:03satisfiable.
3:31:04So f_sub_1 and f_sub_2,
3:31:07yes, it is not satisfiable. So option b
3:31:09is right.
3:31:12So can't it be c like
3:31:16f_sub_1, f_sub_2 are not tology?
3:31:21Neither f_sub_1 see neither f_sub_1 nor
3:31:24f_sub_2 are tology. What is the meaning
3:31:26of it? That means f_sub_1 cannot be
3:31:28true. Neither f_sub_2 can be true. But
3:31:30here in my cases I'm getting f_sub_1 as
3:31:33a true. Here in this cases I'm getting
3:31:35f_sub_2 as a true. So that is why the
3:31:36option c is not right.
3:31:39Okay. Uh any problem in this question?
3:31:43Okay. First try to scan the first and
3:31:46tell me what you're getting that answer.
3:31:49Okay. Scan the first one and tell me you
3:31:51have to use the type one or you have to
3:31:52use the type three. So type one
3:31:54immediately. Type one.
3:31:57Type one. Good. Option B.
3:32:00So, type three. Type three. Type three.
3:32:03Immediately. And that is valid or
3:32:05invalid.
3:32:10So, that is valid.
3:32:13By by seeing it, you can answer that
3:32:14question, right? Yes sir. Yes.
3:32:18Okay. See, option C try to solve. I
3:32:21think rest of this question we don't
3:32:23have to uh find out
3:32:29for rest of this question we don't have
3:32:31to use the type one or type three
3:32:32because they have already given you have
3:32:34to use the type three there is no option
3:32:36C valid
3:32:39option C is valid or invalid valid valid
3:32:43sir valid sir option D okay sol
3:32:48did you solve this question or you
3:32:50solving right Sir, we have solved it.
3:32:53Okay. So, immediately give me the answer
3:32:55first. Sir,
3:32:59G, sir. Sir, G, sir. G, sir. D G G for
3:33:04good.
3:33:06G for good. I think this we have already
3:33:08discussed in the previous question.
3:33:11Is it valid?
3:33:13Yeah, it is valid.
3:33:15This one is P that implies Q that also
3:33:19implies R. This is P or S. This is T
3:33:23that will implies Q and here you'll have
3:33:25S. Right? Now what will happen? For
3:33:28example,
3:33:29uh no S. Huh. Now if you will combine
3:33:34these two, what you will get? You will
3:33:36get P. Now take the first one and P that
3:33:39will implies Q that will implies R. What
3:33:42you will get? You will get Q that will
3:33:43implies R. So P implies Q, Q implies R.
3:33:47If you combine this, you will get T
3:33:48implies R. Take contraositive negation
3:33:50of R that will implies
3:33:55F
3:34:02F F F F F F F F F F F F F F F F F F F F
3:34:02F F F F F F F F F F F F F F F F F F F F
3:34:03F F F F F F F F F F F F F F F F F F F F
3:34:03F Okay, I'll I'll tell you focus you
3:34:07will get immediately
3:34:08take P from here. Simplification P and P
3:34:11you if you'll use you'll give you you'll
3:34:13get R and Q from R and Q you will take
3:34:16simplification R R match you'll get S or
3:34:19T of S you'll get P so it is valid
3:34:24but Q
3:34:28and Q
3:34:31in
3:34:32P and Q huh P and Q you can take P
3:34:36outside simplification
3:34:39when We compared from for for P and Q
3:34:41and P implies R and Q. Can you solve it
3:34:44once? So getting Q.
3:34:48Yes sir. Okay. Okay. I I I'll solve it.
3:34:50But uh did you understand G? So shall I
3:34:54shall I remove it? This one.
3:34:58Yes sir. You can remove sir.
3:35:03Okay. I'll I'll write stepwise step
3:35:05wise. So you will get it. For example,
3:35:08here the P and Q. I'm writing just you
3:35:12know all these things. So this is P and
3:35:15Q which is given.
3:35:18So from this you can write P and then
3:35:20this is simplification.
3:35:23Second primis is P that implies R and Q.
3:35:27So modusonance you will get R and Q.
3:35:30From this you can take R outside.
3:35:33Simplification.
3:35:34Next premises R implies Sority modus you
3:35:39will get Sority negation of S is there
3:35:42and then you'll get T. So it is valid
3:35:45clear is
3:35:48okay see I think we have already
3:35:51discussed right this you will get if you
3:35:55don't understand anything use god's
3:35:57negation of P or Q this is negation of Q
3:36:01one positive one negative you will get
3:36:03negation of P and here you'll have
3:36:05negation of R and then there is a hand
3:36:07condition so you can write negation of P
3:36:09and negation of R take negation outside
3:36:11you'll get P or
3:36:16Clear. Yes sir. Sorry sir.
3:36:21H E.
3:36:24Yes sir.
3:36:27Take P and P. What you'll get? You will
3:36:31get you'll Q implies R will be free.
3:36:35This is use God's rule. Here you'll get
3:36:38negation of P or R. Here also use God's
3:36:41rule. what you'll get you'll get
3:36:44negation I mean you'll get positive Q or
3:36:47negation of P one positive one negative
3:36:50you'll get R or negation P you can use P
3:36:53two times you can use P you will get R
3:36:56did I tell you that we can use P two
3:36:58times three times yes sir I told
3:37:02so for E you can get it understood all
3:37:06this get question
3:37:09get questions are always easy but
3:37:11Because once you will have the habit of
3:37:14solving uh my questions definitely the
3:37:17gate questions will become easier
3:37:222012 solve.
3:37:25Are are you able to see?
3:37:29Yes sir. Yes sir. Okay.
3:37:32Okay. See if you'll see this question.
3:37:36First question. If it rains try to
3:37:38convert it into logical expression. If
3:37:40it rains and do not use your brain while
3:37:44writing while understanding this. For
3:37:47example, let me tell you if you will
3:37:49read this statement and if you try to
3:37:51evaluate by yourself, trust me, you'll
3:37:53forget about if it rains then the
3:37:56cricket match will not be played
3:37:59and then the cricket match was played
3:38:01and then you'll think okay there was no
3:38:02rain. Second read the second if it rains
3:38:04then the cricket match will not be
3:38:06played. It did not rain so the cricket
3:38:08match was played. If you'll read like
3:38:10this, if you'll come up with your own
3:38:12evaluation, 100% both statement will
3:38:15become true. And here comes the beauty
3:38:17of mathematics. If it rains, cricket
3:38:19match will not be played. Negation of P.
3:38:22The cricket match was played. Okay, try
3:38:24to take the contraositive that means P
3:38:27that will implies negation of R, P and P
3:38:30and therefore conclusion. So there was
3:38:33no range. So this will become valid and
3:38:35then it will become very easily we can
3:38:36try to solve it. Second question, if it
3:38:39rains first premises, if it rains in the
3:38:41cricket match will not be played. Fine,
3:38:43it did not rain that means negation of
3:38:45R. Okay, try to take contraositive P
3:38:48implies negation of R. Negation of R and
3:38:51then therefore P. Now as you can see
3:38:54that right side matching right side
3:38:56right side does not imply the left side.
3:38:58So that is why the second one will
3:38:59become invalid. Understood? If you have
3:39:02remember I may have I have explained
3:39:04sometimes may sometimes may not.
3:39:10Yes sir.
3:39:13Understood. Not understood. Tell me
3:39:14first.
3:39:16Understood.
3:39:18Understood. So that's the beauty of
3:39:20mathematics bro. It will help you to
3:39:23solve all the questions.
3:39:25Just a second and I'm giving you all the
3:39:28new questions. Able to solve it? Able to
3:39:30see it? Everyone? Yes sir. Good. So this
3:39:35is one logical question. try to convert
3:39:37this logic into
3:39:39uh variables and then solve this. You
3:39:42can solve immediately. This you can
3:39:44solve immediately. This is something
3:39:45that I have increased the level. So
3:39:48solve it and then we'll discuss about
3:39:50it. Let's close this topic today. Sir,
3:39:53are these three different question or
3:39:55same question? No, this is different.
3:39:58This is today I have given this
3:39:59question.
3:40:03Sir the para question is different from
3:40:05the about two, right?
3:40:07Huh? There are three questions. This is
3:40:09first, second and third.
3:40:11Okay. Okay. Okay. Okay. See, for the
3:40:13first one, if we are writing P double
3:40:16implication Q, this we can write as a
3:40:18negation of P implies Q and Q that will
3:40:23implies negation of P. Right? So here we
3:40:26have a two entities. So we can take one
3:40:29which we requires by using
3:40:30simplification.
3:40:32So if you will use this we have to use P
3:40:35I mean if you use the contraositive
3:40:37negation of R that will implies negation
3:40:39of Q this is negation of R we will get
3:40:41negation of Q. So negation of Q either
3:40:45you can use this or you can use this no
3:40:46problem right suppose if I will be using
3:40:49this by simplification I will take only
3:40:52left part and take contraositive
3:40:54negation of Q that will implies P from
3:40:57there we can take negation of Q and
3:40:59hence we will get P easy
3:41:02sir
3:41:03yes sir
3:41:07okay
3:41:09uh let's try to solve this take P from
3:41:11here match it You will get Q match it
3:41:14QQQ
3:41:16this R or S will be free simplification
3:41:20you can take R one positive one negative
3:41:22you will get negation of T or U
3:41:25simplification you can take T and then
3:41:27you will get U valid
3:41:29yes sir understood or shall I write the
3:41:33whole thing understood
3:41:36everyone please reply
3:41:44If the band could not play rock music.
3:41:48So if that means negation of m music
3:41:53or refreshments were not delivered
3:41:57not delivered
3:42:00then
3:42:02so if the band could not play rock music
3:42:05or the refreshments were not delivered
3:42:08on time right if then the new year's
3:42:12party would have been cancelled
3:42:15Then
3:42:19party cancelled
3:42:22and
3:42:25Alicia would be angry.
3:42:27Angry, right? This will become one
3:42:30statement.
3:42:33If the party were cancelled.
3:42:35So first one negation of M or negation
3:42:39of B. This implies C or A.
3:42:45If the party were cancelled
3:42:48implies then the refunds would have not
3:42:51made. Then the refunds would have to be
3:42:54made. Refunds
3:42:58no refunds were made. So negation of
3:43:00refunds then what we have to then the
3:43:03play music we have to come up with M.
3:43:08Right?
3:43:12Is this the question? Yes sir.
3:43:16Okay. So if you will take
3:43:20contraositive
3:43:21contraositive
3:43:23negation of R implies
3:43:27negation of C. Negation of R you will
3:43:30get negation of C. Take the
3:43:32contraositive here. What you will get?
3:43:34You will get negation of C or A.
3:43:39Am I right?
3:43:42Yes sir. Yes sir. C
3:43:45and
3:43:49right. This is and
3:43:55uh
3:43:57and A. So if you'll take negation of C.
3:44:00Take contraositive for the first
3:44:02statement what you'll get? You'll get
3:44:03negation. You'll get C and A. And then
3:44:06that will implies negation negation of M
3:44:09or negation of D. See we are getting
3:44:12negation of C. Can we write negation of
3:44:15C or negation of A by using addition
3:44:17rule?
3:44:24Why I I'm writing this because I need
3:44:26here. So if you'll write negation of C
3:44:28or negation of A, this will implies M
3:44:31and negation of D. So I will require
3:44:34this. I I I was getting negation of C
3:44:37but here I will require negation of C or
3:44:40negation of A. So by using the addition
3:44:42rule I can write or negation of A. Am I
3:44:45right? Yes sir. Sir by using the true or
3:44:48false. So in in that way you have been
3:44:52taking the addition.
3:44:56No you can use the addition rule here
3:44:58now because see they are telling in the
3:45:01left side this will be negation of C or
3:45:03negation of A. Here we are getting
3:45:05negation of C. If you're getting
3:45:06negation of C and we want this so we can
3:45:10use the addition rule negation of C or
3:45:12negation of A. So you can use it here.
3:45:14Negation of C or negation of A. Left
3:45:17left matches you will get M and negation
3:45:19of D. And what will be my next step?
3:45:23Simplification.
3:45:25Simplification I will require M. Clear
3:45:28everyone?
3:45:30Yes.
3:45:34Okay.

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