Chapter 1

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

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

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

Come from Lewis Carroll's Symbolic Logic. All philosophers are logical. An illogical person is always obstinate. \(\therefore\) Some obstinate persons are not philosophers.

Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: x

Prove each directly. The product of any even integer and any odd integer 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. $$(\exists x)|\mathrm{P}(x) \vee Q(x)|$$

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. $$(\exists x)[\mathrm{P}(x) \vee \mathrm{Q}(x) |$$

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

Prove each directly. The product of any even integer and any odd integer is even.

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

Come from Lewis Carroll's Symbolic Logic.) No ducks waltz. No officers ever decline to waltz. All my poultry are ducks. \(\therefore\) My poultry are not officers.

Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: x

Prove each directly. The square of every integer of the form \(3 k+1\) is also of the same form, where \(k\) is an arbitrary integer.

Lewis Carroll's Symbolic Logic. No ducks waltz. No officers ever decline to waltz. All my poultry are ducks.

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. $$(\forall x)|\mathrm{P}(x) \vee \mathbf{Q}(x)|$$

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

Prove each directly. The square of every integer of the form \(3 k+1\) is also of the same form, where \(k\) is an arbitrary integer.

Give the simplest possible conclusion in each argument. Assume each premise is true. $$\begin{aligned} &p \leftrightarrow q\\\ &\sim p \vee r\\\ &\sim r \end{aligned}$$

Prove each directly. The square of every integer of the form \(4 k+1\) is also of the same form, where \(k\) is an arbitrary integer.

Rewrite each sentence symbolically, where \(\mathrm{P}(x) : x\) is a 16 -bit machine, \(\mathrm{Q}(x) : x\) uses the ASCII" character set, and the UD \(=\) set of all computers. There is a computer that is a \(16-\) bit machine and uses the ASCII character set as well.

Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: xx)]$$

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

Prove each directly. The square of every integer of the form \(4 k+1\) is also of the same form, where \(k\) is an arbitrary integer.

Use De Morgan's laws to evaluate each boolean expression, where \(x=2\) \(y=5,\) and \(z=3\) $$\sim |(x

Give the simplest possible conclusion in each argument. Assume each premise is true. $$\begin{aligned} &p \rightarrow q\\\ &p \vee \sim r\\\ &r \end{aligned}$$

Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: xx)]$$

Prove each directly. The arithmetic mean \(\frac{a+b}{2}\) of any two nonnegative real numbers \(a\) and \(b\) is greater than or equal to their geometric mean \(\sqrt{a b} .\) IHint: consider \((\sqrt{a}-\sqrt{b})^{2} \geq 0.1\)

Rewrite each sentence symbolically, where \(\mathrm{P}(x) : x\) is a 16 -bit machine, \(\mathrm{Q}(x) : x\) uses the ASCII" character set, and the UD \(=\) set of all computers. We can find a 16 -bit computer that does not use the ASCII character set.

Prove each directly. The arithmetic mean \(\frac{a+b}{2}\) of any two nonnegative real numbers \(a\) and \(b\) is greater than or equal to their geometric mean \(\sqrt{a b}\) I Hint: consider \((\sqrt{a}-\sqrt{b})^{2} \geq 0.1\)

Rewrite each sentence symbolically, where \(P(x): x\) is a 16 -bit machine, \(\mathbf{Q}(x): x\) uses the ASCII" character set, and the UD = set of all computers. We can find a 16 -bit computer that does not use the ASCII character set.

Use De Morgan's laws to evaluate each boolean expression, where \(x=2\) \(y=5,\) and \(z=3\) $$\sim[(y

Give the simplest possible conclusion in each argument. Assume each premise is true. $$\begin{aligned} &p \rightarrow \sim q\\\ &\sim r \rightarrow q\\\ &p \end{aligned}$$

Evaluate each boolean expression, where \(a=2, b=3, c=5,\) and \(d=7\). $$[\sim(a>b)] \vee[\sim(c

Prove each using the law of the contrapositive. If the square of an integer is even, then the integer is even.

Rewrite each sentence symbolically, where \(\mathrm{P}(x) : x\) is a 16 -bit machine, \(\mathrm{Q}(x) : x\) uses the ASCII" character set, and the UD \(=\) set of all computers. We can find a computer that is either a \(16-\) bit machine or does not use the ASCII character set.

Prove each using the law of the contrapositive. If the square of an integer is even, then the integer is even.

Rewrite each sentence symbolically, where \(P(x): x\) is a 16 -bit machine, \(\mathbf{Q}(x): x\) uses the ASCII" character set, and the UD = set of all computers. We can find a computer that is either a 16 -bit machine or does not use the ASCII character set.

Use De Morgan's laws to evaluate each boolean expression, where \(x=2\) \(y=5,\) and \(z=3\) $$\sim|(x \geq y) \vee(y

Give the simplest possible conclusion in each argument. Assume each premise is true. $$\begin{aligned} &p \rightarrow q\\\ &\sim r \rightarrow \sim q\\\ &\sim r \end{aligned}$$

Evaluate each boolean expression, where \(a=2, b=3, c=5,\) and \(d=7\). $$[\sim(b

Prove each using the law of the contrapositive. If the square of an integer is odd, then the integer is odd.

Rewrite each sentence symbolically, where \(\mathrm{P}(x) : x\) is a 16 -bit machine, \(\mathrm{Q}(x) : x\) uses the ASCII" character set, and the UD \(=\) set of all computers. There exists a computer that is neither a 16 -bit machine nor uses the ASCIl character set.

Prove each using the law of the contrapositive. If the square of an integer is odd, then the integer is odd.

Use De Morgan's laws to evaluate each boolean expression, where \(x=2\) \(y=5,\) and \(z=3\) $$\sim|(x

The program is running if and only if the computer is working. The computer is working or the power is off. The power is on.

Evaluate each boolean expression, where \(a=2, b=3, c=5,\) and \(d=7\). $$\sim[(a>b) \vee(b \leq d)]$$

Prove each using the law of the contrapositive. If the product of two integers is even, then at least one of them must be an even integer.

Prove each using the law of the contrapositive. If the product of two integers is even, then at least one of them must be an even integer.

Evaluate each boolean expression, where \(a=2, b=3, c=5,\) and \(d=7\). $$\sim\\{(a \leq b) \wedge[\sim(c>d)]\\}$$