Math 2534 Solution Homework 2 Spring 2017

Transcription

1 Math 2534 Solution Homework 2 Spring 2017 Put all work on another sheet of paper unless told otherwise. Staple multiple sheets. Problem 1: Use Algebra of Logic to prove the following and justify each step. [( p q) p] [ ( p q)] p Proof : [( p q) p] [ ( p q)] given [( p q) p] [ ( p q)] Implication law [( p q) p] [ p q)] DeMorgan's law [( p q) p] [ p q)] Double Negative law p [ p q)] Absorption Law p Absorption Law Therefore [( p q) p] [ ( p q)] p B) [ p ( q r)] ( q p) p ( q r) Proof: [ p ( q r)] ( q p) given [ p ( q r)] ( q p) Implication Law [( p q) ( p r)] ( q p) Distriputive Law [( p q) ( q p)] ( p r) Commutative and associative Law [( p q) ( p q)] ( p r) Commutative Law [( p q)] ( p r) Idempotent Law [( p ( q r) Distributive Law Therefore [ p ( q r)] ( q p) p ( q r)

2 Problem 2: Using algebra of logic put the following in Disjunctive Normal Form ( p q) [( p r) ( q r)] ive a reason to validate each step below. ( p q) [( p r) ( q r)] iven ( p q) [ ( p r) ( q r)] Implication Law ( p q) [( p r) ( q r)] DeMorgan's Law ( p q) ( p) [( r) ( q r)] Associative Law ( p q) ( p) [( r q) ( r r)] Distributive Law ( p q) ( p) [( r q) T] Negation Law ( p q) ( p) ( r q) Identity Law ( p q) ( p p) ( r r) ( q q) Idempotent Law Problem 3: Put the following into symbolic implication form. Define all your variables. a) I will clean up only if you help. Define C to be the statement I will clean up Define H to the statement you help. C H b) ame will be postponed since it is raining. Define to be the statement ame is postponed Define R to the statement It is raining. R c) I will not go to the movie or I will not study. Define M to be the statement I will go to the movie Define S to the statement I will study. M S M S

3 Problem 4: If it snows then we will go skiing. Solution: Let S be the statement: It snows Let K be the statement: We go skiing. So S K 1) Rewrite the above sentence in inverse form. S K If It does not snow, we will not go skiing. 2) Rewrite the above sentence in converse form. K S If we go skiing, then it did snow. 3) Rewrite the above sentence in contrapositive form. K S If we did not go skiing, then it did not snow. Problem 5: Determine if the following arguments are valid and justify your conclusion. Put each argument into symbolic logic and define all variables. In justifying your conclusion be sure to indicate what is the sufficient condition and what is the necessary condition. a) If you are in the Marching Virginians, then you must go to the game. You went to the game. Therefore you are in the Marching Virginians. Define B to be the statement you are in the Marching Virginias Define to be the statement You go to the game. B B Since we are given the necessary condition, nothing can be determined about B. This is converse error and an invalid argument. b) If the test is Thursday, you will miss the game. You did not miss the game. Therefore you did not have a test. Define T to be the statement the test is Thursday. Define to be the statement You will miss the game. T T This argument is valid by the contrapositive.

4 Problem 6: P,Q and R represent the following statements: P: Jim is a CS Major Q Anne is an EE Major R Laura is an Environmental Science Major M Charlie is a Math Major Assume that the expression (P R ) ( Q M) is false and that P is false and M is true. Put the following statements into implication form and determine if the sufficient and necessary conditions are true or false and if the implication is true or false. a) Anne is a EE Major or Charlie is not a Math Major. b) Jim is a CS Major and Anne is not an EE Major. c) Only if Anne is a EE Major is Jim a CS Major Solution: Since the implication (P R ) ( Q M) is given to be false, the necessary condition Q M must be false while the sufficient condition P R is true. It is also given that P is false and therefore R is true in order for the disjunction to be true and R is false. We also know that M is true and for the conjunction to be false, Q is false. d) Anne is a EE Major or Charlie is not a Math Major. Q M F T F F F e) Jim is a CS Major and Anne is not an EE Major. P Q F F F T F f) Only if Anne is a EE Major is Jim a CS Major P Q P Q F F T F T

5 Problem 7: Below is logic puzzle by Lewis Carroll. In these puzzles he strings together a list of implications and the job of the reader is to use all the listed implications to arrive at an inescapable conclusion. Put all statements into symbolic implication form. Determine the conclusion and justify your reasoning using sentences. 1) Promise breakers are untrustworthy 2) Wine drinkers are very communicative. 3) A man who keeps his promises is honest. 4) No teetotalers are pawnbrokers. 5) One can always trust a very communicative person. Use the following Variables for you symbolic argument so that we all have the same notation. P: Keeps Promises T: Is Trustworthy W: Drinks Wine C: Very communicative H: Is honest A: Is a pawnbroker Solutions: 1) P T 2) W C 3) P H 4) W A 5) C T Using the contrapositive we have that P T T P and W A A W Using the argument form of transitive we have that A W C T P H Therefore A H so Pawnbrokers are honest.