Math 2534 Solution Homework 2 sec

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1 Math 2534 Solution Homework 2 sec Problem 1: Use Algebra of Logic to Prove the following: [( p q) ( p q)] heorem: [( p q) ( p q)] Pr oof : [( p q) ( p q)] q given [( p q) ( p q)] q Implication Law [ ( p q) ( p q)] q DeMorgan's Law [( p q) ( p q)] q DeMorgan's Law and Double Neg Law [( p q] q Idempotent Law q [( p q) ( p q)] Identity Law Problem 2: Put the following into implication form. Define all your variables. Solution: Define P to be the statement you will pass. Define S to be the statement you study. Define to be the statement you take driving test. Define k to be the statement you get a skate board. a) You will pass this test only if you study. P S b) You will pass this test if you study. S P c) ake the driving test or get a skate board. K K

2 Problem 3: Are any the following statements equivalent? Put into symbolic logic and justify your reasoning. Solution: Define D to be the statement you drive. Define K to be the statement you drink. a) Only If you drive, do not drink. K D b) Drive if you do not drink. K D c) Do not drink or do not drive. K D K D Statement a) is equivalent to b) since the sufficient and necessary conditions match. Problem 4: Given the statement: If it does not rain then Laura will rock climb. Let R be the statement: It rains Let L be the statement Laura will rock climb. Original statement is R L Rewrite this sentence as directed below: a) Inverse form : R L If it rains then Laura will not climb. b) Converse form: L R If Laura climbs then it will not rain. c) Contrapostive form: L R If Laura did not climb then it rained. d) Contradiction form: ( R L) R L It did not rain and Laura did not climb. Problem 5: Determine if the following arguments are valid and justify your conclusion. a) Jane will go to the concert only if Bill goes. Bill does not go to the concert. herefore Jane did not go. Solution: Let J be the statement: Jane will go to the concert. Let B be the statement: Bill will go to the concert. J B B J Valid argument: his is the equivalent contrapositive form.

3 b) If you miss class, you will not do well. You did not miss class. herefore you did well. Solution: Let M be the statement: You miss class. Let W be the statement: You do well. M M W W Invalid argument: his is the Inverse error and not equivalent to the original Conditional statement. c) If it is hot we will go swimming. We did go swimming herefore there it was hot. Solution: Let H be the statement: It is hot Let S be the statement: We will swim. H S S H Invalid argument: S is the necessary condition and does not guarantee the sufficient condition. his is converse error. Problem 6: Murder at the Bate s Motels It was a dark night, heavy with rain and wind and very late. hree lone travelers, strangers to each other, chanced to meet in front of the Bate s Motel. hey were suspicious of each other, but they approached the motel to seek refuge for the night. A sour faced receptionist opened the door and said the owner was sleeping but he could find them each rooms. hey were given clean towels and shown to their rooms.

4 During the night a murder was committed. he crime can be considered somewhat unusual. he identity of the culprit is unknown, but the identity of the victim is also unknown. he only possible choices for either is the receptionist, the owner (Mr. Bates), and each of the three travelers. Given the following clues, determine the culprit and the victim. Put the following statements into symbolic logic and explain your reasoning. 1) If the traveler in room 1 was the culprit, then the traveler in room 3 was the victim. 2) If the traveler in room 2 was the victim, then the receptionist was the culprit. 3) If the traveler in room 3 was the victim, then traveler in room 2 was the culprit. 4) If the receptionist was the culprit, the victim was the traveler in room 3. 5) he receptionist was not available until the next morning, and was not able to provide an alibi. 6) If the traveler in room 3 was the culprit, the traveler in room 2 was the victim. 7) If the traveler in room 2 was the culprit, then the receptionist is the victim. Use the following notation where stands for traveler, C stands for culprit, V stands for victim, R stands for receptionist and B for owner. Let 1, 2 and 3 represent the rooms. 1C 2C 3C 1V 2V 3V R R C V Solution: 1) 2) 3) 4) R 1c 2v 5) R 6) 7) 2c c 3c 2v 2c R v c R v

5 We know that the receptionist was not the victim since he showed up the next morning. herefore the statement that the receptionist is the victim is false. his statement is the necessary condition in statement 7 and since it is false but the implication is true, the sufficient condition is also false. herefore we know that traveler 2 was not the culprit. his statement (that traveler 2 was the culprit) is false and also the necessary condition for the implication in statement 3. By the same reasoning as before, we know that traveler 3 is not the victim. he statement (traveler 3 is the victim) is false and is the necessary condition in both statements 1 and 4. Since these implication statements are true, the sufficient conditions are false. So neither the receptionist nor traveler 1 is the culprit. By continuing to use the same reasoning we have by statement 2 that traveler 2 is not the victim and by statement 6 we determine that traveler 3 is not the culprit. herefore traveler 1 must be the victim and the owner, Mr. Bates, must be the culprit.