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CmSc 175 Discrete Mathematics Homework 03: Logic, Arguments due date: 02/10/11 1. Using the equivalence laws to show that ~(~(P Q) V P ) V Q Q (15 pts) 2. Write the contrapositive, converse, inverse and negation of the following expressions. Apply De Morgan’s laws if necessary: (12 pts) a. P Q V R Contrapositive: Converse: Inverse: Negation: b. P R Q Contrapositive: Converse: Inverse: Negation: 3. Assuming that the premises are true, determine whether the arguments are valid or invalid. If valid, indicate the inference rule. If invalid, indicate the type of the error: (8 pts) 3.1. Premises: a. If I go to the movies, I won't finish my homework b. If I don't finish my homework, I won't do well on the exam tomorrow. Conclusion: If I go to the movies, I won't do well on the exam tomorrow. Valid Invalid 1 3.2. Premises: a. If John is not playing, the team will not win. b. If the team does not win, the trip will be postponed. c. John is playing. Conclusion: The trip is not postponed Valid Invalid 4. Show how we can prove the truth of U from the premises below. Indicate the inference rule at each step in the proof. (20 pts) (1) P ~Q (2) R ~S (3) ~P S U (4) ~R Q (5) S For example, the first step in the proof would be: (6) ~R, by (2), (5) and Modus Tollens. Now we can add (6) to the premises, and then combine (6) and (4) and apply MP to conclude (7) Q The idea is to look for a pair of premises that matches an inference rule, and then apply that rule to make a new conclusion. See all inference rules in the table in Lesson 04. 5. For each of the sentences below: Represent the following sentences as quantified logical expressions using the predicates : student(x), soccer_player(x), and healthy(x). (We can assume that the domain is ‘all people’ and we do not indicate it explicitly) Find the negation of the logical form Example: Students that play soccer are healthy. Expression: x, student (x) soccer_player(x), healthy(x). Negation: x, student(x) soccer_player(x) ~ healthy(x). (18 pts) 2 a. All healthy students play soccer Expression: Negation: b. Some healthy soccer players are students Expression: Negation: c. No soccer players are healthy students Expression: Negation: 6. Indicate whether the following arguments in predicate logic are valid or invalid (underline the correct answer). If valid, indicate the inference rule. If invalid, indicate the type of the error: (12 pts) 6.1. 6.2. 6.3. All teachers grade homework Peter grades homework Therefore peter is a teacher Valid Invalid Valid Invalid Valid Invalid No lions like cherries Peter likes cherries Therefore Peter is not a lion Pilots are happy but not wealthy Peter is wealthy Therefore Peter is not a pilot 7. Prove that the sum of two odd numbers is even (15 pts) 3