91Ó°ÊÓ

Let \(p, q, r\) denote primitive statements. a) Use truth tables to verify the following logical equivalences. i) \(p \rightarrow(q \wedge r) \Leftrightarrow(p \rightarrow q) \wedge(p \rightarrow r)\) ii) \([(p \vee q) \rightarrow r] \Leftrightarrow[(p \rightarrow r) \wedge(q \rightarrow r)]\) iii) \([p \rightarrow(q \vee r)] \Leftrightarrow[\neg r \rightarrow(p \rightarrow q)]\)

Verify the first Absorption Law by means of a truth table. }

Use truth tables to verify that each of the following is a logical implication. a) \([(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)\) b) \([(p \rightarrow q) \wedge \neg q] \rightarrow \neg p\) c) \([(p \vee q) \wedge \neg p] \rightarrow q\) d) \([(p \rightarrow r) \wedge(q \rightarrow r)] \rightarrow[(p \vee q) \rightarrow r]\)

\begin{aligned} &\text { Let } p, q, r, s \text { denote the following statements: } p: 1 \text { finish writing my computer program } \\ &\text { before lunch; } q \text { : I shall play tennis in the afternoon; } r: \text { The sun is shining; } s \text { The } \\ &\text { humidity is low. Write the following in symbolic form. } \\ &\text { a) If the sun is shining, I shall play tennis this afternoon. } \\ &\text { b) Finishing the writing of my computer program before lunch is necessary for my } \\ &\text { playing tennis this afternoon. } \\ &\text { e) Low humidity and sunshine are sufficient for me to play tennis this afternoon. } \end{aligned}

Let \(p, q, r\) denote the following statements about a particular triangle \(A B C, p:\) Triangle \(A B C\) is isosceles; \(q\) : Triangle \(A B C\) is equilateral; \(r\); Triangle \(A B C\) is equiangularTranslate each of the following into an English sentence. a) \(q \rightarrow p\) b) \(\neg p \rightarrow \neg q\) c) \(q \leftrightarrow r\) d) \(p \wedge \neg q\) e) \(r \rightarrow p\)

Determine the truth value of each of the following implications. a) If \(3+4=12\), then \(3+2=6\). b) If \(3+3=6\), then \(3+6=9\). c) If \(3+3=6\), then \(3+4=9\). d) If Thomas Jefferson was the third president of the United States, then \(2+3=5 .\)

Let \(p, q, r\) denote primitive statements. Write the converse, inverse, and contrapositive of each of the following implications. a) \(p \rightarrow(q \wedge r)\) b) \((p \vee q) \rightarrow r\)

Rewrite each of the following statements as an implication in the If-Then form. a) Practicing her serve daily is a sufficient condition for Darci to have a good chance of winning the tennis tournament. b) Fix my air conditioner or I won't pay the rent. c) Mary will be allowed on Larry's motorcycle only if she wears her heimet.

Construct a truth table for each of the following compound statements, where \(p, q, r\) denote primitive statements. a) \(\neg(p \vee \neg q) \rightarrow \neg p\) b) \(p \rightarrow(q \rightarrow r)\) c) \((p \rightarrow q) \rightarrow r\) d) \((p \rightarrow q) \rightarrow(q \rightarrow p)\) e) \([p \wedge(p \rightarrow q)] \rightarrow q\) f) \((p \wedge q) \rightarrow p\) g) \(q \leftrightarrow(\neg p \vee \neg q)\) h) \([(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)\)

For the universe of all integers, let \(p(x), q(x), r(x), s(x)\), and \(t(x)\) be the following open statements. $$ \begin{array}{ll} p(x): & x>0 \\ q(x): & x \text { is even } \\ r(x): & x \text { is a perfect square } \\ s(x): & x \text { is (exactly) divisible by } 4 \\ t(x): & x \text { is (exactly) divisible by } 5 \end{array} $$ a) Write the following statements in symbolic form. i) At least one integer is even. ii) There exists a positive integer that is even. iii) If \(x\) is even, then \(x\) is not divisible by \(5 .\) iv) No even integer is divisible by \(5 .\) v) There exists an even integer divisible by \(5 .\) vi) If \(x\) is even and \(x\) is a perfect square, then \(x\) is divisible by \(4 .\) b) Determine whether each of the six statements in part (a) is true or false. For each false statement, provide a counterexample. e) Express each of the following symbolic representations in words. i) \(\forall x[r(x) \rightarrow p(x)]\) ii) \(\forall x[s(x) \rightarrow q(x)]\) iii) \(\forall x[s(x) \rightarrow \neg t(x)]\) iv) \(\exists x[s(x) \wedge \neg r(x)]\) v) \(\forall x[\neg r(x) \vee \neg q(x) \vee s(x)]\) d) Provide a counterexample for each false statement in part (c).