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CHAPTER 2: MATHEMATICAL REASONING, PROBLEM SOLVING AND LOGIC — Transcript

by Engr. Trexie Arugay · 4,980 words · 927 segments · language en · Watch on YouTube

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0:10So good day everyone. Let's start our
0:13lecture. So nao for chapter 2
0:17mathematical reasoning problem solving
0:19and logic. So for learning outcome
0:24distinguish inductive from deductive
0:26reasoning and use each appropriately.
0:30Explain the roles of intuition
0:34counter example proofs and certainty and
0:37mathematics.
0:38Apply polia fourstep strategy to
0:41structured engineering problems.
0:44Then identify simple and compound
0:47statement and use common logical
0:50operations.
0:52Construct truth table for negation,
0:55conjunction, disjunction, conditional
0:57and by conditional statements. And last
1:00for our learning outcome, uh evaluate
1:03argument validity and build elementary
1:06formal proofs using valid rules of
1:10inference.
1:11So let's start with 2.1.
1:15Um let's compare inductive and deductive
1:18reasoning.
1:20Wait lang.
1:22So when we say inductive reasoning, it
1:25moves from a particular observation
1:28toward a general pattern or conjecture.
1:32While
1:34um for deductive reasoning, it moves
1:37from accepted premises, definition,
1:40actions or uh even previously
1:43established results toward a conclusion
1:47that logically follows.
1:50So let's compare inductive and deductive
1:54reasoning using our diagram.
1:57So
2:00when we have observed examples or data
2:05we will use uh general conjecture. So it
2:09is applicable for inductive reasoning
2:14while um if we use known rules or
2:18actions it will now apply the deductive
2:23reasoning.
2:26Since inductive reasoning is particular
2:29observation toward a general pattern or
2:33conjecture,
2:34every time we use a pattern or
2:37conjecture, we uh we define this as
2:41inductive reasoning. That's why in our
2:45visual representation
2:47since we use general conjecture uh we
2:50labeled it as inductive reasoning.
2:54So now we'll go to deductive reasoning
2:58for observed examples or data.
3:02We can identify if um the sample is
3:07deductive reasoning when we encounter
3:11>> [snorts]
3:12>> uh known rules, actions or even uh
3:15previously established results.
3:22Then after the known rules or actions,
3:26it will now be labeled as deductive
3:29reasoning.
3:31So for the last part or
3:34um after evaluating if it's inductive or
3:38deductive reasoning now we will arrive
3:43identifying it as proof or test
3:46depending on the uh examples or data
3:49given to us.
3:53So for a clearer view we have here uh
3:56reasoning type we have two the first one
3:59is the inductive then the next one is
4:02the deductive. So inductive uh direction
4:06let's start with the direction first. So
4:09we have specific cases that includes
4:14um general conjecture
4:16while uh for deductive
4:19we have general rule plus what are the
4:23premises
4:25then uh followed by a specific
4:28conclusions.
4:33So for inductive reasoning strength we
4:37have suggestive not automatically
4:40certain. So
4:42we apply suggestive for inductive.
4:47When uh we encounter deductive
4:51there are uh certain
4:54if the premises are true and if the
4:57reasoning is valid. So meaning uh when
5:01we use deductive
5:03the example or the data should be true
5:08and uh it should also have a valid
5:12reasoning.
5:15So for typical use inductive is
5:18typically used in pattern discovery or
5:23uh even in empirical modeling
5:27while deductive reasoning is typically
5:30used as proof verification and uh
5:34rulebased design since uh based on its
5:37strength
5:39the premises are true and the reasoning
5:42should be valid.
5:47So now uh as computer engineering uh
5:51students we have here
5:53sample wherein you can
5:57see the difference between inductive and
5:59deductive reasoning. So we have computer
6:02engineering example here for inductive
6:04reasoning uh after testing many packet
6:08sizes infer that the latency tends to
6:12increase with congestion.
6:15For deductive example
6:18if every rising clock edge trigger
6:22a register update and a rising edge
6:25occurs then the register update occurs.
6:29So as we can see for inductive
6:33um for inductive example we have here
6:37suggestive and pattern discovery
6:40approach while for our deductive example
6:45um we can uh say that the premises
6:51are true and the reasoning is valid. So
6:56that's why it is labeled as deductive
6:59example.
7:01So let's try our work example 2.1
7:06for inductive pattern.
7:09So for the problem observe the outputs
7:12of a counter 1 3 57 9. Predict the next
7:17output and state the limitation of your
7:19reasoning.
7:21So for step one um we have here the
7:25differences between consecutive terms
7:28are all + two. So meaning
7:32uh select
7:34we have the outputs of counter
7:381 + 2
7:42is equals to 3. So
7:462 + 3 is equals to 5.
7:51Then
7:555 + 2 is equals to 7.
8:037 + 2 is equals to 9.
8:10So for the outputs of the counter
8:14we
8:15have the difference uh differences
8:18between consecutive terms.
8:21So add + 2 to get the next output of
8:26counter. We started at 1 + 2 then 3. So
8:322 + 3 then 5. So 5 + 2 again 7 output.
8:39Then last output n counter is 7 + 2
8:42which is 9.
8:45So since
8:47step one, we will now proceed to step
8:50two. The plausible rule is
8:54start one then add two every time.
9:01Next term. So in this case
9:059 last output of counter. So 9 + 2
9:12is equals 11. So 11 e to y magig next
9:16term nin.
9:19So for step three uh however ainite list
9:23can fit many rules without additional
9:25information.
9:2711 is a uh reasonable conjecture but not
9:31logically forced. So as a result or as
9:36interpretation
9:38inductive reasoning is useful for
9:40discovering pattern but not uh as a
9:45additional evidence or a rule is needed
9:48for certainty.
9:52So that's why we use inductive reasoning
9:54here
9:56cuz uh we are trying to discover the
9:59pattern of the uh we are trying to
10:02discover the pattern of outputs for the
10:07uh respected counter.
10:11So now let's try a work example 2.2
10:16for deductive reasoning.
10:18So we have here the problem premise one
10:23is if reset is equals to 1 then output Q
10:27is equals to zero then for premise 2
10:30reset is equals to 1. So what follows?
10:35So let's um do the first step
10:40represent the premises as P to Q and P.
10:45So now let's apply the rule of inference
10:48or the modus ponent. So therefore uh if
10:53we have q q is equals to zero based on
10:58our uh premise.
11:04So as a conclusion or as the result
11:07follows it follows uh deductively from
11:11the two premises that we have. So
11:14premise one wherein if reset is equals
11:17to one then output Q is zero. Then for
11:19premise two if reset is equals to 1 then
11:23Q is zero.
11:30Now I added here um additional
11:34um inductive reasoning to predict the
11:37next number in each sequence. Let's zoom
11:39this one first.
11:43So for letter A we have the sequence 3 6
11:479 12 15.
11:50So the sequence is increasing three each
11:55time. So meaning
11:58so we started at I sorry bumalik so we
12:03started at three. So to get the next
12:06term what we'll do is
12:09[snorts]
12:10um three
12:13+ 3 is equ= to 6.
12:17Now um 6 + 3 is = 9.
12:24Then 9 + 3 is =
12:2812.
12:3212 + 3 is equals to 15.
12:38So the next number for this sequence
12:41would be 18 cuz 15 + 3 is equals to 18.
12:49So another example
12:53um
12:55for
12:57letter B we have here the
13:03sequence
13:04that follows an increasing pattern.
13:10Wait lang e ano k lang y
13:16increasing by three. This one is
13:19increasing pattern.
13:21So to do this
13:26the sequence will be 2 3 4 5 6. So
13:31meaning
13:33um 1 +
13:362 is equ= to 3.
13:42Then 3
13:44+ 3 is equals to 6.
13:51Then 6 + 4 is equ= to 10.
14:01Now 10
14:03+ 5 is equals to
14:0815.
14:10Then
14:1215 + 6 we will get 21.
14:17So we have increasing pattern for this
14:20letter B.
14:26Now let's have the example two. Use
14:30inductive reasoning to make a
14:32conjecture. So first we will pick a
14:35number and we'll multiply the number by
14:378.
14:39then add six to the product. Then let's
14:42divide the sum by two and subtract
14:44three.
14:46So for this one, we will use inductive
14:48reasoning to make a conjecture about the
14:52relationship between size of resulting
14:54number and the size of the original
14:57number. So conjecture ginagamit is
15:01inductive reasoning.
15:04So let's start.
15:08So based on the procedure start with
15:11five then multiply down by 8. So 5 * 8
15:15is 40. Now let's add 6 to the
15:21uh 40. So we will now have 46. Then
15:25let's divide the 46 by 2. We will get
15:2823. Then let's subtract three
15:33on the 23. Now we will have 20.
15:38So for um for another example multiply
15:44by 8. So 10 * 8 is 80. Let's add 6 to
15:4880. We will now have 86.
15:51So 86 / 2 would be 43. So for the last
15:57step
15:59you 3 4 we will now get 40. So as you
16:04can see we are observing the uh pattern
16:09of pick a number then multiply the
16:12number by 8 then add six to the product
16:14then divide the sum by two and subtract
16:18three and we use here
16:21inductive reasoning. So same as this
16:25one.
16:29So
16:30now let's try another example.
16:45So it [snorts]
16:47example number three use inductive
16:50reasoning to solve and application.
16:57So A if a if a pendulum has length of 49
17:01units what is its period? Then for B if
17:06the length of pendulum is quadrupled
17:08what happens to its period.
17:12So we will use this length of pend
17:16pendulum in units and period of pendulum
17:20in heartbeat. So 1 uh for length 1 4 9
17:2616 25 and 36 for the beats or the heart
17:31beats 1 2 3 4 5 6.
17:36So
17:41we have here length square root and
17:44period. Let's calculate the square root
17:47of each length and compare it to the
17:50period.
17:52So as we can see here the square root of
17:55length is equals to the period of each
17:58data point. So
18:01uh square root
18:03um you square root is equal s period of
18:07each data point. So uh after 36 we will
18:11now have 49. So square root n 7 49. So
18:16on period is 7. So e to y newly added
18:21nin data
18:25since the square root of the length is
18:28equal then s period of each data point.
18:32So meaning if I'm square root s
18:37length one
18:41I sorry so length one
18:47you period is period
18:52one. So ganyan
18:56this why um seven length is 49. So
19:03since y
19:07lang square root
19:11is 7 edging equal to 7 din period kagan
19:17to
19:19equal to m to
19:24so ganun.
19:25Okay,
19:31now let's go to deductive reasoning.
19:38So for deductive reasoning, show that
19:41the following procedure produces a
19:43number that is four times the original
19:45number. Then for procedure, pick a
19:48number, multiply the number by eight,
19:50then um add six to the product, divide
19:53sum by two, then subtract three.
19:57So for deductive reasoning, we have the
20:00steps. Pick number, multiply number by
20:048, add six to the product, divide the
20:06sum by two, then subtract three.
20:15So conclusion your final result is 4x
20:20which is four times the original number.
20:24Therefore the procedure consistently
20:27produce a number that is four times from
20:30uh from its original number regardless
20:33of the chosen number.
20:47So inductive versus deductive reasoning.
20:50Example number five.
20:55Determine deductive or inductive but a
20:58and bin.
21:00So s a during the past 10 years a tree
21:04has produced plums every other year. So
21:08last year the tree did not produce
21:11plums. So this year the tree will
21:14produce plums.
21:16So observation
21:19is inductive.
21:23So argument a or a
21:27based observation tree
21:31plums every other year for 10 years. So
21:34conclusion
21:37uh this year the tree will produce plums
21:40cuz last year produce plums. So uh type
21:44of reasoning is inductive cuz
21:48the argument is based on past
21:50observation and uh it makes a prediction
21:54about the future.
21:56So bas
22:00all home improvements cost more than the
22:03estimate. The contractor estimated that
22:06my home improvement will cost 35,000.
22:09Thus my home improvement will cost more
22:12than 35,000.
22:14So general rule apply sabi. So all home
22:20improvements cost more than the
22:22estimate.
22:24So specific case the contractor
22:26estimated the 35,000.
22:29So as conclusion
22:31the home improvement will cost more than
22:3435,000.
22:35So this type of reasoning is deductive
22:39since um this argument applies a general
22:44rule to the specific situation to reach
22:47a conclusion.
22:51So here we have um
22:551 2 3 4 5 and six now um conclusion. So
23:02identify deductive or inductive
23:05reasoning ba?
23:09So Andrea noticed that every Saturday
23:12her neighbor moans his lawn. Today is
23:15Saturday. So Andrea concludes her
23:17neighbor will mow his loan.
23:21So argument one observation notice
23:24Andrea
23:28loan every Saturday. So conclusion since
23:32today is Saturday
23:40um neighbor. So this type of reasoning
23:43is inductive
23:47argument based on s past observation and
23:50the predict about the future.
23:53For number two, students at Blake's high
23:56school must have a B average in order to
23:59participate in sports. So Blake has a B
24:03average. So he concludes that he can
24:06participate in sports at school.
24:09So for this one, uh, argument two,
24:13general rule, students at Blakes's High
24:15School must have a B average to
24:19participate in sport. So specific case
24:22for this one since Blake has a B average
24:27conclusion
24:29s Blake participate for the school event
24:34or Dun sport. So type of reasoning is
24:37deductive
24:39since this argument applies uh applies
24:42the general rule to a specific situation
24:45to reach a conclusion.
24:48So, same goes with argument three. For
24:51example, number three.
24:54At Ora school, if you are late five
24:56times, you will receive a detention.
24:59Oria has been late for school five
25:01times, therefore he will receive
25:03detention. So, argument uh let's start
25:06with general rule. If you're late five
25:09times, you will receive detention.
25:11Specific case, um Oraha has been late
25:14five times. So, for conclusion,
25:18Oria will receive a detention. So type
25:21of argument
25:23uh is deductive since it applies general
25:26rule to a specific conclusion uh sorry
25:30specific situation to reach a
25:32conclusion.
25:34So eto argument four, five and six um
25:38last two deductive apply general rule to
25:42a specific situation while you four is
25:46inductive since
25:48uh as we can see here um your argument
25:52is based on s past observation and
25:55prediction about future.
25:59So next tan.
26:03So eto example six solve a logic puzzle.
26:11So each of four neighbors Shan, Maria,
26:15um Sara and Brian has different
26:18occupation, editor, banker, chef or
26:21dentist. So from the following clues
26:23provided, let's determine the occupation
26:26of each neighbor. So
26:30um let's proceed with the solution.
26:34Maria gets home from work after the
26:37bunker but before the dentist. So si
26:40Maria
26:42um
26:43since
26:46occupation
26:49possible banker possible dentist. So for
26:53this one possible naang is editor or
26:58chef. So
27:02sis Sara
27:04is the one
27:06last
27:08and editor. So sis Sara
27:13editor. So you dentist
27:17and sis Sara live for work at the same
27:20time. So since um Sara
27:25editor and wedding dentist so chef or
27:32banker
27:35while si Maria
27:38banker dentist then
27:44for the last clue the banker lives next
27:47door to Brian. So si Brian Hindi shop
27:49wedding mager.
27:52So
27:55yan
27:58first step
28:00since
28:03Maria chakasara. So analyze
28:08si Maria
28:14is possible chef
28:18chef
28:20Maria chef
28:23option for Sara is banker while Brian
28:32Banker
28:34ayun then since wala na y option n
28:39chef and bunker.
28:42So
28:44and
28:46possible editor dentist say Brian
28:50then C Shan is editor.
28:56So na solve
28:58logic puzzle.
29:01So let's try example seven.
29:06So here um the sequence is 2 6 12 20 30
29:10then n raised to 2 + n to get the next
29:14term. So a sub 1 is 2, a sub 2 is 6, a
29:19sub3 is 12, a sub4 is 20, a sub5 is 30.
29:23to get the next term. Uh, a subn is
29:27equals to n to 2 + n.
29:35So, we have um 2 6 12 20 30
29:462 is A1.
29:52Next term a n is equ= n to 2 + n to get
30:00the next term.
30:08So now uh try nin difference table.
30:12So sequence is 2 58
30:16114.
30:18sequence.
30:22Um
30:28three sila.
30:31So magali po 3.
30:36So try
30:392 + 3 is equals to 5.
30:47Okay. Wait
30:52five.
30:53Um
30:575 + 3 is equals to 8. So 8
31:06+ 3 is equals to
31:1011.
31:12Then
31:1511.
31:16Next is 11 +
31:203 is equals to 14.
31:24So 14.
31:27So now we have the difference table.
31:31So same with example number eight. So we
31:35have the first difference then second
31:37difference. Soap. Second difference of
31:41this table is by four.
31:47Uh 5 + 9 is equ= to 14 14 + 13 is equals
31:52to 27. 27 + 17 44. So 44 + 21 is 65. So
32:03second difference
32:07number of difference. So 4a gamit. So 4
32:12uh 9 + 4 is 13. [music]
32:1513 + 4 is 17. 17 + 4 is 21.
32:26So now uh let's go to
32:31proof versus experiment. So we have here
32:34different types of claim we have the
32:36universal.
32:37So for every X meron tong P parenthesis
32:42X. So this proves a generally or
32:47disprove with one counter example. P
32:50evaluate
32:52for ex uh for existential nan there
32:56exist x such that p parenthesis x. So
33:01for conditional if p then q. So this
33:05implies a direct proof contraositive or
33:09contradiction.
33:13Now let's proceed to 2.3 the polyia four
33:17step in problem solving.
33:22So first step is understand the problem.
33:24Second is devise a plan. Uh third step
33:27carry out the plan. Then fourth step is
33:30look back and improve.
33:34So these are the key questions for each
33:38step. So for step one understand the
33:40problem. what is known, what is unknown,
33:44what are the units, constraints and
33:46assumptions.
33:47Soak
33:49to understand the problem for step one.
33:52Uh for step two key questions would be
33:55which example, model, uh formula,
33:58pattern, table, graph or simpler case
34:02can help? So parap
34:05plan. Then for step three, carry out the
34:08plan. Can each step be justified and
34:12check?
34:14So last is step four. You key questions.
34:19Is the answer reasonable? Can it uh can
34:22it be verified another way? What changes
34:26if assumptions change? So naman question
34:30itat for step four.
34:39So we have here um three example.
34:47Let's apply the solution. Ah sorry
34:50problem. Problem one. Baseball team wins
34:54and losses.
34:56Losses. Sorry. Understand the problem.
34:59We need to find the number of different
35:01orders in which a baseball team can have
35:03two wins and two losses in four games.
35:08So solution since it's a combination
35:12problem we need to find the number of
35:13ways to choose two games out of four to
35:17be wins. So formula
35:21uh in this case you n is 4
35:26n is equ= to 4 y total games total
35:31games.
35:34Then rin is 2 which is your number
35:39of wins.
35:46So combinational uh s discrete math
35:51combinational
35:53c 2 is equals to 4
35:57divide s 2
36:014 - 2
36:05then t s
36:08uh 24 / 4 is equals to 6.
36:17So for example number two teenagers
36:21ages. So we need to find the ages of
36:24three teenagers whose uh product is 4590
36:30and who are all different ages. So
36:34[music]
36:34what we'll do here is we start by prime
36:37factoring.
36:41So 4590 is equals to
36:45going up
36:474590
36:49is equals to
36:512 * 3 * 3 * 3 I sorry * 5 nto * 17.
37:00Now um what we'll do is we need to group
37:05these factors into three groups. So it
37:09represent nin by ages.
37:12So t three groups. So three groups.
37:18So group one, G1, G1 for group one, G2
37:22for group two and G3
37:25for group three. So we have uh 2 * 3 is
37:32equals to 6.
37:34Now we also have um 3 * 5 is equals to
37:3915. Then last is 17. So now we arrive at
37:44the [snorts] ages of teenagers
37:48which is 6, 15 and 17.
38:01So problem three uh eto mangtomas goats
38:06and ducks. So understand the problem. So
38:09megaas goats and ducks. So counting
38:12heads are 39 counting legs are 110
38:19or isolve is
38:22goats and ducks.
38:26So let's use G for goat. Goat. Let's use
38:32D for ducks.
38:37So equation based on s given
38:40information.
38:43So G + D
38:47is 39. So buy goat plus ducks. Tong 39
38:54individuals.
38:56Uh since part n counting heads is 39
39:04to y labelled as counting heads 39 based
39:08on s given
39:10uh 4 g + 2 d would be equal to 110
39:21legs to is heads.
39:25So is solve equation first um uh for the
39:30first equation
39:34is
39:36multiply the first equation by two and
39:39subtract it from the second equation.
39:43So
39:462
39:48goat is equals to 32. So goat is 16.
39:54applying the um first equation.
40:00So for
40:02uh this one substituting g is equal to
40:0616 into first equation. So since we have
40:09here 16
40:12we will use it to uh know how many dots
40:16are there. So 16 + b is equals to 39. So
40:22d is equals to 23.
40:25So
40:27239
40:29total number
40:31minus na y number n goat which is 16.
40:36Now we will get 9 - 6 so 23
40:4223.
40:44So ay number of
40:48ducks and goat nin. So for goat we have
40:5216 then for duck we have 23.
41:04So example number four
41:07each one ana Alvin and Johnny have
41:11different favorite color among red blue
41:14green and orange. So no person's name
41:17contains the same number of letters as
41:20his her favorite color. So Albin and the
41:23boy who likes blue lives in different
41:26parts of town. Then red is the favorite
41:29color now one of the girls. So it's
41:32either you see an orya.
41:35What is each person's favorite color?
41:39So apply nin.
41:43So apply nin based lang s clue. No
41:45person's name contains the same number
41:48of letter as his favorite color.
41:52Okay,
41:54this means an cannot like red or blue.
42:00Anna cannot like green or orange. Alvin
42:03cannot like blue or green. Then Johnny
42:05cannot like red and orange. So [snorts]
42:07for step two, Alvin and the boy who
42:09likes blue lives in different parts of
42:11town. This means Alvin does not like
42:15blue. Then Alvin cannot like blue or
42:19green. So um
42:23baka red favorite color
42:26blue or green. So red ya red is the
42:30favorite color of one of the girls. So
42:35since we have here an in and India as a
42:39girl um we conclude that an likes color
42:44red.
42:45Then Alvin also likes red. So for Enya
42:51the uh the other girl he cannot like
42:54green or blue. So due to the length of
42:58uh rule so she must like blue.
43:03So dito sa conclude blue y favorite
43:07color. So Johnny is left with green
43:12silang hindi occupied na color. So as
43:16conclusion
43:18um therefore
43:20the favorite colors are for an color red
43:24for Anna color blue for Alvin also red
43:27for Johnny color green.
43:35So punt s 2.4
43:41wherein um we states here a statement or
43:47proposition is declarative sentence is
43:50either true or false
43:53but not uh both under a given
43:55interpretation. So ped um
44:01true ba or false in preposition
44:06to mga simple statement.
44:09So cut is English
44:12sentence
44:14two is math
44:17expression.
44:19The word cat begins with k. So English
44:22sentence
44:25uh e to math
44:29expression and so on.
44:36So the answer for this 1 to 15
44:42is
44:44so cut is false that this is not a
44:47sentence but a noun. Okay pala en pala
44:52dapa to so en cas stands for English
44:57noun en
45:03two is false cuz it is a number not
45:06sentence the word cat begins with k the
45:09word cat begins with c so false 1 + 2 is
45:13equal to 4 false cuz 1 + 2 is 3
45:17so 5 - 3 is also false in complete
45:21mathematical expression. So 5 - 3
45:26= to 2 true true the cat is black true x
45:30is false since it is not complete
45:33sentence. So x = 1 true x - 1 is = to 0
45:39true uh t + 3 false t + 3 is equals to 3
45:45+ t so true since uh ng paradoxial.
45:50So x +
45:54z
45:56x then hot sat but is false.
46:02So for uh compound statement and logical
46:06operations
46:08uh we have the basic the three basic the
46:12not
46:14the or and the and
46:19so
46:21uh we will arrive at end
46:24if the condition through only when both
46:28inputs are true. So for or um
46:33true when at least one input is true
46:36since addition approach then for end we
46:39have multiplication
46:42approach. So not is the reverse
46:47for truth table.
46:53So we have uh different kind of
46:56operation. we have the negation or when
46:59we read not P for conjunction P and Q.
47:05So for disjunction P or Q for
47:08conditional if P then Q. Then for by
47:12conditional P if and only if Q soap
47:16negation
47:18um
47:21true when P is false. So
47:25input I1 negate
47:29zero. So conjunction
47:32if two inputs are true
47:36condition. So input one is one input
47:40two is one. So one input one is one and
47:48input I zero mag uh for false yan. So
47:54this junction input one is one or input
47:59two is
48:01zero. So one pen one or one one pen
48:10false lang
48:12input uh one and two par zero.
48:22So familiar naman s table.
48:26So negation and or uh conditional by
48:32conditional. So we have here
48:35uh variables P and Q. So for our truth
48:39table we have
48:42T T FF. So for variable Q alternate T
48:47FTF.
48:48So
48:56original value. So from t nag f from t
49:00nag f from f n t from f n t. So e to uh
49:07end
49:10e to is or
49:12tapos e to
49:15um wait lang conditional cha by
49:18conditional
49:21conditional and by conditional.
49:26So kap and
49:29approach is multiplication. So e to
49:32addition. So ta inputs P and Q. So T * T
49:39is equals to T. T * F is F. F * T is F.
49:44F * F is F. So for this one
49:48um
49:50T + T is T. T + F is T. F + T is T. If
49:58false. So for conditional.
50:02So P and P to Q. So
50:07T
50:09appos
50:11F T.
50:16So by conditional nam man T FT.
50:26So familiar
50:28table
50:33example.
50:35So we also have the logical equivalent.
50:38So double negation
50:41uh deorgans
50:43and conditional and contraositive. So
50:47equivalent form
50:49double negation
50:51porans
50:55conditional and contraositive.
51:01So double negation means um negating
51:04twice. So it will return to its original
51:08statement or original value.
51:11So the Morgan one
51:13not of n becomes not um not of n becomes
51:18or of not. So for the morgan 2 n man
51:24not of or becomes
51:28not of or becomes
51:31end of not. S Morgan one
51:36not of end becomes or of not.
51:43Soap negate twice
51:47original form or original value.
51:52So conditional
51:55not P or Qap
52:01contraositive
52:06conditional I equivalent contraositive
52:09value which is
52:15Q S P.
52:25modus tolins and modus ponins. Then we
52:29also have the hypothetical
52:32syllogism and disjunction syllogism.
52:37So e to additional form or rule of
52:40inference lanto.
52:45So we have the direct proof,
52:48contraositive, contradiction and cases
52:52formal proofs
52:54direct proof
52:59hypothesis and uh logically derive
53:02conclusion. So for contraositive
53:05so
53:07not Q to not P instead of P to Q. So ya
53:15for contradiction assume desired
53:18conclusion is false and uh derive
53:22impossibility
53:24for cases divide the problem into um
53:27exhaustive possibilities solve.
53:35So example d uh direct proof prove if an
53:38integer is n
53:41if an integer n is divisible by 8 then n
53:46is even. So step one assume that n is
53:49divisible by 8. So by definition n is
53:53equals to 8k. So for some integer k. So
53:57step two we'll write n. So 8kos
54:03number. So two parentheses 4 k. Then for
54:08step three because 4k is integer then n
54:12has the form 2 m for integer m. So kayan
54:16equals t
54:18um 4k. So therefore n is an even by
54:23definition.
54:25So the proof does not rely on examples
54:28such as 8, 16 or 24.
54:32Then apply tag
54:36apply every integer
54:39divisible by 8 satisfy or direct proof.
54:50So [music]
54:522.10
54:54to you feature
54:57for inductive and deductive reasoning.
54:59So starting point typical direction,
55:01strength of conclusion and CPE example
55:05stated
55:07partina
55:09inductive reasoning may specific
55:11observation
55:13example measurement or cases deductive
55:16naman accepted facts premises or
55:19previously proved now result. So
55:24uh inductive reasoning specific cases
55:27then
55:28conjecture or general pattern. So
55:31deductive reasoning general rule
55:34logically necessary
55:40case.
55:41So deductive my valid reasoning form to
55:45while s inductive
55:50counter example.
55:53So
55:54CPA example
55:59so deductive general rule and uh
56:03logically
56:06conclusion for a case. So deductive bin
56:10accepted facts, premises, actions or
56:13previously
56:15uh proved results. So for inductive ulit
56:18specific observation, examples,
56:20measurement or cases.
56:26So summary. So
56:31is understand the problem, deise a plan,
56:34uh carry out the plan. Then fourth is
56:36look back.
56:40So prepositional logic me negation,
56:43conjunction, disjunction, exclusive or
56:46implication and by conditional.
56:50Um to symbolic form so not P and Q, P or
56:55Q, P X or Q, P implication to Q and P by
57:00conditional to Q. So
57:04sila mag true.
57:09So for truth table
57:16truth table.
57:19So eto formal proofs pattern summary. So
57:24summary
57:26discuss.
57:28So that ends our chapter 2
57:32uh lecture for
57:34MMW. So balikas.
57:39So
57:41chapter 2 mathematical reasoning problem
57:44solving and logic.
57:46So learning outcomes
57:53inductive from deductive.
57:57Ano ba
58:00counter example proofs and certainty in
58:04mathematics
58:06apply poly fourstep strategy
58:11identify simple and compound statement
58:17logical operations
58:19and or not and so
58:26table negation, conjunction,
58:28disjunction, conditional and by
58:31conditional statement. Then last is you
58:34proofs or you formal proofs
58:39using valid rules of inference. So
58:42[music]
58:43ion end chapter 2
58:47>> [music]

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