Chapter 1

Rewrite each implication in inferential form. $$|(p \rightarrow q) \wedge(\sim q)| \rightarrow \sim p$$

Which of the following are propositions? The earth is flat.

Determine the truth value of each proposition, where the UD consists of the numbers \(\pm 1,\pm 2,\) and \(0 .\) $$(\exists x)\left(x^{3}+2 x^{2}=x+2\right)$$

Rewrite each implication in inferential form. $$|(p \rightarrow q) \wedge(q \rightarrow r)| \rightarrow(p \rightarrow r)$$

Which of the following are propositions? Toronto is the capital of Canada.

Give the truth value of \(p\) in each case. \(p \equiv q, q \equiv r,\) and \(r\) is true.

Determine if each implication is trivially true. If \(n\) is a prime number, then \(n^{2}+n\) is an even integer.

Determine if each implication is trivially true. If \(n\) is a prime number, then \(n^{2}+n\) is an even integer.

Verify that each inference rule is a tautology. $$p \rightarrow(p \vee q)$$

Verify each, where \(f\) denotes a contradiction. $$\sim(\sim p) \equiv p$$

Verify each, where \(f\) denotes a contradiction. (See Table \(1.14 . )\) $$ p \wedge p \equiv p $$

Determine if each implication is trivially true. If \(n \geq 41,\) then \(n^{3}-n\) is divisible by 3

Verify that each inference rule is a tautology. $$[(p \rightarrow q) \wedge(q \rightarrow r) | \rightarrow(p \rightarrow r)$$

Verify each, where \(f\) denotes a contradiction. $$p \wedge p \equiv p$$

Verify each, where \(f\) denotes a contradiction. (See Table \(1.14 . )\) $$ p \vee p=p $$

Prove each directly. The sum of any two even integers is even.

Test the validity of each argument. $$p \vee q$$ $$q \vee r$$ $$\sim r$$ $$\overline{\therefore p}$$

Determine the truth value of each proposition, where the UD consists of the numbers \(\pm 1,\pm 2,\) and \(0 .\) $$\sim(\forall x)\left(x^{3}=x\right)$$

Find the truth value of each compound statement. $$(5<8) \text { and }(2+3=4)$$

Test the validity of each argument. $$\begin{aligned} &\begin{array}{l} p \vee q \\ q \vee r \\ \sim r \\ \hline \end{array}\\\ &\therefore p \end{aligned}$$

Verify each, where \(f\) denotes a contradiction. $$p \vee p \equiv p$$

Prove each directly. The sum of any two even integers is even.

Prove each directly. The sum of any two odd integers is even.

Test the validity of each argument. $$p \leftrightarrow q$$ $$\sim p \vee r$$ $$\sim r$$ $$ \overline {\therefore \sim q}$$

Find the truth value of each compound statement. Paris is in France or \(2+3=4\)

Determine the truth value of each proposition, where the UD consists of the numbers \(\pm 1,\pm 2,\) and \(0 .\) $$(\forall x)\left[\sim\left(x^{5}=4 x\right)\right]$$

Test the validity of each argument. $$\begin{aligned} &p \leftrightarrow q\\\ &\begin{array}{l} \sim p \vee r \\ \sim r \\ \therefore \sim q \end{array} \end{aligned}$$

Verify each, where \(f\) denotes a contradiction. $$p \wedge q \equiv q \wedge p$$

Prove each directly. The sum of any two odd integers is even.

Verify each, where \(f\) denotes a contradiction. (See Table \(1.14 . )\) $$ p \vee q \equiv q \vee p $$

If Bill likes cats, he dislikes dogs. Bill likes dogs.

Prove each directly. The square of an even integer is even.

Find the truth value of each compound statement. If \(1=2,\) then \(3=3\).

Verify each, where \(f\) denotes a contradiction. $$p \vee q \equiv q \vee p$$

Prove each directly. The square of an even integer is even.

Verify each, where \(f\) denotes a contradiction. (See Table \(1.14 . )\) $$ \sim(p \vee q) \equiv \sim p \wedge \sim q $$

If Pat passes this course, she will graduate this year. Pat does not pass this course.

Prove each directly. The product of any two even integers is even.

Let \(P(x) : x^{2} > x, Q(x) : x^{2}=x,\) and the UD \(=\) set of integers. Determine the truth value of each proposition. $$(\exists x)|\sim P(x)|$$

Find the truth value of each compound statement. \(\triangle \mathrm{ABC}\) is equilateral if and only if it is equiangular.

Let \(\mathrm{P}(x): x^{2} > x, \mathrm{Q}(x): x^{2}=x,\) and the UD \(=\) set of integers. Determine the truth value of each proposition. $$(\exists x)[\sim \mathbf{P}(x)]$$

Verify each, where \(f\) denotes a contradiction. $$\sim(p \vee q) \equiv \sim p \wedge \sim q$$

Prove each directly. The product of any two even integers is even.

Verify each, where \(f\) denotes a contradiction. (See Table \(1.14 . )\) $$ \sim(p \rightarrow q)=p \wedge \neg q $$

Frank bought a personal computer or a video cassette recorder (VCR). If he bought a VCR, then he likes to watch movies at home. He does not like to watch movies at home.

Prove each directly. The square of an odd integer is odd.

Let \(P(x) : x^{2} > x, Q(x) : x^{2}=x,\) and the UD \(=\) set of integers. Determine the truth value of each proposition. $$(\exists x)|\mathrm{P}(x) \wedge \mathrm{Q}(x)|$$

Verify each, where \(f\) denotes a contradiction. $$\sim(p \rightarrow q) \equiv p \wedge \sim q$$

Prove each directly. The square of an odd integer is odd.

If Peter is married, he is happy. If he is happy, then he does not read the computer magazine. He does read the computer magazine.