91Ó°ÊÓ

Write the converse and contrapositive of each of the following conditional statements. (a) If \(a=5,\) then \(a^{2}=25\). (b) If it is not raining, then Laura is playing golf. (c) If \(a \neq b,\) then \(a^{4} \neq b^{4}\). (d) If \(a\) is an odd integer, then \(3 a\) is an odd integer.

Suppose that Daisy says, "If it does not rain, then I will play golf." Later in the day you come to know that it did rain but Daisy still played golf. Was Daisy's statement true or false? Support your conclusion.

For each of the following, use a counterexample to show that the statement is false. Then write the negation of the statement in English, without using symbols for quantifiers. * (a) \((\forall m \in \mathbb{Z})\left(m^{2}\right.\) is even \() .\) (b) \((\forall x \in \mathbb{R})\left(x^{2}>0\right)\) (c) For each real number \(x, \sqrt{x} \in \mathbb{R}\) (d) \((\forall m \in \mathbb{Z})\left(\frac{m}{3} \in \mathbb{Z}\right)\). (e) \((\forall a \in \mathbb{Z})\left(\sqrt{a^{2}}=a\right)\). "(f) \((\forall x \in \mathbb{R})\left(\tan ^{2} x+1=\sec ^{2} x\right)\)

For each of the following statements \- Write the statement as an English sentence that does not use the symbols for quantifiers. \- Write the negation of the statement in symbolic form in which the negation symbol is not used. \- Write a useful negation of the statement in an English sentence that does not use the symbols for quantifiers. (a) \((\exists x \in \mathbb{Q})(x>\sqrt{2})\) (b) \((\forall x \in \mathbb{Q})\left(x^{2}-2 \neq 0\right)\). * (c) \((\forall x \in \mathbb{Z})(x\) is even or \(x\) is odd). (d) \(\left(\exists x \in\right.\) Q) \((\sqrt{2}

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of "I will play golf and I will mow the lawn" as "I will not play golf or I will not mow the lawn." (a) We will win the first game and we will win the second game. (b) They will lose the first game or they will lose the second game. (c) If you mow the lawn, then I will pay you \$20. (d) If we do not win the first game, then we will not play a second game. (e) I will wash the car or I will mow the lawn. (f) If you graduate from college, then you will get a job or you will go to graduate school. (g) If I play tennis, then I will wash the car or I will do the dishes. (h) If you clean your room or do the dishes, then you can go to see a movie. (i) It is warm outside and if it does not rain, then I will play golf.

Suppose that \(P\) and \(Q\) are statements for which \(Q\) is false and \(\neg P \rightarrow Q\) is true (and it is not known if \(R\) is true or false). What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(\neg Q \rightarrow P\) (b) \(P\) (c) \(P \wedge R\) (d) \(R \rightarrow \neg P\)

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers \(\mathbb{Z}\). (a) \(n+7=4\). (b) \(n^{2}=64\). (c) \(\sqrt{n} \in \mathbb{N}\) and \(n\) is less than 50 . (d) \(n\) is an odd integer that is greater than 2 and less than 14 . (e) \(n\) is an even integer that is greater than 10 .

Construct a truth table for each of the following statements: (a) \(P \rightarrow Q\) (c) \(\neg P \rightarrow \neg Q\) (b) \(Q \rightarrow P\) (d) \(\neg Q \rightarrow \neg P\) Do any of these statements have the same truth table?

Use truth tables to prove each of the distributive laws from Theorem 2.8 . (a) \(P \vee(Q \wedge R) \equiv(P \vee Q) \wedge(P \vee R)\) (b) \(P \wedge(Q \vee R) \equiv(P \wedge Q) \vee(P \wedge R)\)

Use set builder notation to specify the following sets: (a) The set of all integers greater than or equal to 5. (b) The set of all even integers. \(\star(c)\) The set of all positive rational numbers. (d) The set of all real numbers greater than 1 and less than 7 . \star (e) The set of all real numbers whose square is greater than 10 .