2345_2019-Spring-HW2-Proofs-Solutions.pdf - Math 2345 \u2013 Discrete Mathematics Homework 2 Methods of Proof Solutions Due Date Monday(beginning of class

# 2345_2019-Spring-HW2-Proofs-Solutions.pdf - Math 2345...

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Homework 2: Methods of Proof SolutionsMath 2345 – Discrete MathematicsHomework 2: Methods of Proof SolutionsDue Date:Monday, February 04, 2019 (beginning of class)Complete the following problems. Show all work and explain your reasoning carefully where appropriate. Justifyyour arguments and supply evidence to back up your claims.Problem 2.1 (Deductions).Tommy Flanagan was telling you what he ate yesterday afternoon. He tells you,“I had popcorn or raisins. Also, if I had cucumber sandwiches, then I drank soda. But I didn’t drink soda or tea.”Of course you know that Tommy is the world’s worst liar, and everything he says is false. What did Tommy eat?Justify your answer by writing all of Tommy’s statements using symbolic variables (P, Q, R, S, T), taking theirnegations, and using these to deduce what Tommy actually ate.Solution 2.1.LetP:I had popcorn. LetQ:I had raisins.LetR:I had cucumber sandwiches. LetS:I dranksoda.LetT:I drank tea.Using the usual intended English meaning, Tommy’s statements symbolically are:PQ,RS,¬(ST)Since all three statements are false, we know that the following are all true:¬P∧ ¬Q,R∧ ¬S,ST.So we know that Tommy did not eat popcorn, he did not eat raisins, and he did not drink soda. Therefore, heate cucumber sandwiches and drank tea.Problem 2.2 (Argument Validity).Decide if the following arguments are valid or invalid:(a)PQ, PR, P(QR)(b)(PQ)R,¬P∨ ¬Q,¬RSolution 2.2.(a) The table below shows that whenever both premises are true, the conclusion is also true (see first five rowsfrom top). This means the argument is valid.PQRQRPremise 1z}|{PQPremise 2z}|{PRConclusionz}|{P(QR)TTTTTTTFTTTTTTFTFFTTTFFTFTTTFFFFTTTTTFFTFFTFTFFTFTFFFFFF(b) Consider the case thatPis false,Qis true, andRis true. Then both premises(PQ)Rand¬P∨ ¬Qare true. However, the conclusion¬Ris false. Therefore the argument is invalid.Problem 2.3 (Trivial and Vacuous Proofs).(a) Letxandybe real numbers. Prove that ifx2+y2<2xy, thenx+y <0.(b) Prove for every real numberxthat ifx3-5x-10, then(x-1)(x-3)≥ -2.