91Ó°ÊÓ

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

7\. For the universe of all integers, let \(p(x), q(x), r(x), s(x)\), nd \(t(x)\) be the following open statements. \(s(x) ; \quad x\) is (exactly) divisible by 4 \(t(x): \quad x\) is (exactly) divisible by 5 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 falsc. For cach false statement, provide a counterexample. c) 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) \(\exists x[s(x) \wedge \neg r(x)]\) d) Provide a counterexample for each false statement in part (c)

7\. a) If \(p, q\) are primitive statements, prove that $$ (\neg p \vee q) \wedge(p \wedge(p \wedge q)) \Leftrightarrow(p \wedge q) $$ b) Write the dual of the logical equivalence in part (a).

8\. 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) \([(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)\)

8\. Give the reasons for the steps verifying the following argument. \((\neg p \vee q) \rightarrow r\) \(r \rightarrow(s \vee t)\) \(\quad \neg s \wedge \neg u\) \(\frac{\neg u \rightarrow \neg t} \\\\{\therefore p} \end{array\) Steps Reasons 1) \(\neg s \wedge \neg u\) 2) \(\neg u\) 3) \(\neg u \rightarrow \neg t\) 4) \(\neg t\) 5) \(\neg s\) 6) \(\neg s \wedge \neg t\) 7) \(r \rightarrow(s \vee t)\) 8) \(\neg(s \vee t) \rightarrow \neg r\) 9) \((\neg s \wedge \neg t) \rightarrow \neg r\) 10) \(\neg r\) 11) \((\neg p \vee q) \rightarrow r\) 12) \(\neg r \rightarrow \neg(\neg p \vee q)\) 13) \(\neg r \rightarrow(p \wedge \neg q)\) 14) \(p \wedge \neg q\) 15) \(\therefore p\)

9\. Write the converse, inverse, and contrapositive of each of the following implications. For each implication, determine its truth value as well as the truth values of its corresponding converse, inverse, and contrapositive. a) If \(0+0=0\), then \(1+1=1\). b) If \(-1<3\) and \(3+7=10\), then \(\sin \left(\frac{3 \pi}{2}\right)=-1\).

10\. Determine whether each of the following is true or false. Here \(p, q\) are arbitrary statements. a) An equivalent way to express the converse of " \(p\) is sufficient for \(q\) " is " \(p\) is necessary for \(q\)." b) An equivalent way to express the inverse of " \(p\) is necessary for \(q\) " is " \(\neg q\) is sufficient for \(\neg p\)." c) An equivalent way to express the contrapositive of \(" p\) is necessary for \(q\) " is " \(\neg q\) is necessary for \(\neg p\)."

12\. Write each of the following arguments in symbolic form. Then establish the validity of the argument or give a counterexample to show that it is invalid. a) If Rochelle gets the supervisor's position and works hard, then she'll get a raise. If she gets the raise, then she'll buy a new car. She has not purchased a new car. Therefore either Rochelle did not get the supervisor's position or she did not work hard. b) If Dominic goes to the racetrack, then Helen will be mad. If Ralph plays cards all night, then Carmela will be mad. If either Helen or Carmela gets mad, then Veronica (their attorney) will be notified. Veronica has not heard from either of these two clients. Consequently, Dominic didn't make it to the racetrack and Ralph didn't play cards all night. c) If there is a chance of rain or her red headband is missing, then Lois will not mow her lawn. Whenever the temperature is over \(80^{\circ} \mathrm{F}\), there is no chance for rain. Today the temperature is \(85^{\circ} \mathrm{F}\) and Lois is wearing her red headband. Therefore (sometime today) Lois will mow her lawn.

12\. a) Let \(p(x, y)\) denote the open statement " \(x\) divides \(y\)," where the universe for each of the variables \(x, y\) comprises all integers. (In this context "divides" means "exactly divides" or "divides evenly.") Determine the truth value of each of the following statements; if a quantified statement is false, provide an explanation or a counterexample. i) \(p(3,7)\) ii) \(p(3,27)\) iii) \(\forall y p(1, y)\) iv) \(\forall x p(x, 0)\) v) \(\forall x p(x, x)\) vi) \(\forall y \exists x p(x, y)\) vii) \(\exists y \forall x p(x, y)\) viii) \(\forall x \forall y[(p(x, y) \wedge p(y, x)) \rightarrow(x=y)]\) b) Determine which of the eight statements in part (a) will change in truth value if the universe for each of the variables \(x, y\) were restricted to just the positive integers. c) Determine the truth value of each of the following statements. If the statement is false, provide an explanation or a counterexample. [The universe for each of \(x, y\) is as in part (b).] i) \(\forall x \exists y p(x, y)\) ii) \(\forall y \exists x p(x, y)\) iii) \(\exists x \forall y p(x, y)\) iv) \(\exists y \forall x p(x, y)\)

13\. Consider the open statement $$ p(x, y): \quad y-x=y+x^{2} $$ where the universe for each of the variables \(x, y\) comprises all integers. Determine the truth value for each of the following statements. a) \(p(0,0)\) b) \(p(1,1)\) c) \(p(0,1)\) d) \(\forall y p(0, y)\) e) \(\exists y p(\mathbb{1}, y)\) f) \(\forall x \exists y p(x, y)\) g) \(\exists y \forall x p(x, y)\) h) \(\forall y \exists x p(x, y)\)