Lab 2 solutions - MATH*2000 Fall 2012 Lab#2 Thursday September 20 Problem 0.1 Test the validity of the following argument If I study then I will not

Lab 2 solutions - MATH*2000 Fall 2012 Lab#2 Thursday...

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MATH*2000 Fall 2012 – Lab #2 Thursday, September 20 Problem 0.1.Test the validity of the following argument:If I study, then I will not fail Math 2000.If I do not play X-box games, then I will study.But I failed Math 2000.Therefore I must have not played X-box games.Proof.LetS= I study,M= I fail Math 2000,X= I play X-box games. Then the aboveargument is as follows1.S⇒ ¬M2.¬XS3.M————————-4.¬XNote that, the contrapositive of (1) isM⇒ ¬S. So, together with (3), we have¬S(thatis,¬Sis true).The contrapositive of (2) is¬SX. Since we know¬Sis true, we haveXis true. But(4) says¬Xis true. SinceX∧ ¬Xis false, the argument is not valid.Problem 0.2.Using the laws of the algebra of propositions and the rules of inference provethatPR,RQ,¬(PQ)¬Pis a valid argument.
2 9. ¬ P ( FALSE ) (from 8 by Complement law) 10. ¬ P (from 9 by Identity law) Problem 0.3. Simplify: 9}. Note we’re dealing with Natural numbers, so the possible solutionsto2x <9are1,2,3,4. Thus the answer is{1,2,3,4}.3.{xZ|xZ}(you won’t be able to list all the elements! just write a few in orderto see a pattern, and then “...”).{,1,4,9,16,25,36, . . .}. Note no negative integersare in this set.4.{xR|x2=x}={,1}. Problem 0.4.Indicate “true” or “false” in each of the following assertions.Explain.1.{a, b} ∈ {{a, b, c},{a, c}, a, b}.2.{a, b} ⊂ {{a, b, c},{a, c}, a, b}.3.∅ ∈ {{a, b, c},{a, c}, a, b}.4.∅ ⊂ {{a, b, c},{a, c}, a, b}.Proof. 1. False. The element{a, b}does not appear in the given set. 2. True:a, b∈ {a, b} ⇒a, b∈ {{a, b, c},{a, c}, a, b}, so by definition,{a, b} ⊂ {{a, b, c},{a, c}, a, b}.3. False. does not appear as an element in the given set.4. True. In class we saw the empty set is always a subset of any set.

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