11.1 The Logic of Making conclusions

Transcription

1 11.1 The Logic of Making conclusions Mar 21 10:12 AM 1

2 Postulates, definitions and Theorems. Dec 5 12:51 PM 2

3 Each justification for a conclusion is a true conditional with the conclusion as the consequence conclusion justification p m t 2 1 m 1 = m 2 // line => corr. 's = if // line impies corr. = then m 1 = m 2 Dec 5 12:52 PM 3

4 4) Law of Detachment Dec 5 12:52 PM 4

5 5) Another way to think of the Law of Detachment q is detached from p => q It is a generalization of the common sense idea that when p => q is a true conditional and p is true, then q also must be true. Dec 5 12:53 PM 5

6 6) Law of Transitivity Dec 5 12:52 PM 6

7 m: measure is between 0 and 90 degrees a: angle is acute m => a and m X is an acute angle Nov 28 4:25 PM 7

8 7) Study Example 1 8) Now try the following problem: p: go to camp b: buy a computer a: make an A a => p p => b Law of Transitivity or Law of Detachment a => b If you make an A, then we will buy you a computer Dec 5 12:53 PM 8

9 9) Study Example 2 10) Now try the following problem: p: Perp. Bis of the bases of an isos trap is symmetry line of an isos. trap. i: a figure is an isos Trap r: a figure is a rectangle (1) r => i (3) i => p (4) If a figure is a rectangle then the perp. bis. of the bases is a sym. line of the rectangle and (2) ABCD is a rectanlge The perp. bis. of the bases of ABCD is the sym. line of ABCD Dec 5 12:53 PM 9

10 about the dish p: pudding n: nice w: wholesome 1) If a dish is pudding then it is nice p => n 3) If a dish is nice then the dish is not wholesome n => not w from 1) and 3) by the Law of Transitivity { p => not w p the dish is pudding by Law of Detachment the dish is not wholesome p => not w Dec 5 12:52 PM 10

11 If a person enters before May 1st then the person meets the deadline. m = enter by May 1st d = meet deadline m => d and m Jack met the deadline Dec 5 12:52 PM 11

12 about people Law of Transitivity or Law of Detachment If a little knowledge is dangerous, then no one is out of danger L: a little knowledge is dangerous N: no one is out of danger L => N and L No one is out of danger. Dec 5 12:52 PM 12

13 5t 11 = 24 5t = 35 Addition Property of Equality t = 7 Multiplication Property of Equality If 5t 11 = 24 then 5t = 35 If 5t = 35 then t = 7 If 5t 11 =24 then t 7 Dec 5 12:53 PM 13

14 RECALL: Hypothesis / Antecedent: Conclusion / Consequent: "if" part of a conditional "then" part of a conditional Instance of the conditional: Both "if" & "then" parts are true! Counterexample: Keep"if" part true, make "then" part false! Mar 21 10:27 AM 14

15 1. Conclusions are derived by common sense and logic. We make conclusions daily without even thinking about it. Mar 21 10:33 AM 15

16 Law of Detachment From a true conditional p=>q and a statement or given information that p is true, you may conclude q. Dec 4 10:20 PM 16

17 2. If p implies q and if p is true, what do you know about q? q IS TRUE! notation "implies" Mar 21 10:35 AM 17

18 Law of Detachment: Mar 21 10:38 AM 18

19 Law of Transitivity: If p implies q and q implies r then p implies r. notation If and then Mar 21 10:43 AM 19

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21 If you want to be successful, you must work hard. If you work hard, you must make sacrifices. What conclusion can you make from this? If you want to be successful, you must make sacrifices. Mar 21 10:45 AM 21

22 Consider this statement: If the moon is made of green cheese, then the sun is made of salsa. Is this a true statement? NO! Is it possible to show that this is an instance? Why/why not? No, because neither the antecedent nor consequent are true! Mar 21 10:46 AM 22

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24 What must we conclude about a statement when the hypothesis is not true? The statement (conditional) will not be true. Mar 21 10:49 AM 24

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28 Assume this statement is true: If a tennis player has won Wimbledon, then the player is world class. Using the Law of Detachment only, what (if anything) can you conclude if you also know the given statement is true? a. Steffi Graf won Wimbledon. b. Pete Sampras has won Wimbledon. c. Anna Kournikova has not won Wimbledon. d. Arthur Ashe has won the U.S. Open Tennis Tournament. Mar 21 10:50 AM 28

29 In some states if a person has a driver s license, then the person s age is greater than or equal to 16. What can you conclude in those states (if anything) if you know the following. a.tom has a driver s license. b. Mary is 18 years old. c. Carrie drives a car legally. d. Billy does not drive a car. Mar 21 10:50 AM 29

30 Determine the final conclusion. Determine if it s Law of Detachment or Law of Transitivity. 1. You want to be successful, so you must study. When you study you take notes. 2. Pizzas are good. Meat lovers are a pizza. 3. All students at AHS love Geometry Class. Becky Homecie is an AHS student. Mar 21 10:54 AM 30

31 Chapter 11 WS A's pdf Dec 4 10:33 PM 31

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33 If the day begins with the letter T then Maria practices the cello. Today begins with T If Jaime pitches for Friday's game, they will win the championship The Cougars will win the championship Dec 9 12:14 PM 33

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35 Chapter 11 WS A's KEY.pdf Dec 4 10:35 PM 35

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58 WSB 11 1 and 11 2.pdf Oct 24 7:07 AM 58

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61 Attachments Chapter 11 WS A's pdf Chapter 11 WS A's KEY.pdf WSB 11 1 and 11 2.pdf