Chapter 1

Prove each using the law of the contrapositive. If the product of two integers is odd, then both must be odd integers.

Negate each proposition, where \(x\) is an arbitrary integer. $$(\exists x)\left(x^{2} \neq 5 x-6\right)$$

Prove each using the law of the contrapositive. If the product of two integers is odd, then both must be odd integers.

Carol is a baby if and only if she is illogical. Either she is illogical or unhappy. But she is happy.

Let \(t\) be a tautology and \(p\) an arbitrary proposition. Give the truth value of each proposition. $$\sim p \vee t$$

Three persons took a room for \(\$ 30\) at a hotel. Soon after they checked out, the room clerk realized she had overcharged them since the room rents for \(\$ 25 .\) She sent a bellhop to them with a \(\$ 5\) reimbursement, but he returned to them only \(\$ 3,\) keeping \(\$ 2\) for himself. Thus the room \(\cos t \$ 30-\$ 3=\$ 27\) and \(\$ 27+\$ 2=\$ 29,\) so what happened to the extra dollar?

Negate each proposition, where \(x\) is an arbitrary integer. There are no white elephants.

Aaron, Benjamin, Cindy, and Daphne are all friends. They are 34,29 \(28,\) and 27 years old, not necessarily in that order. Cindy is married to the oldest person. Aaron is older than Cindy, but younger than Daphne. Who is married to whom and how old are they? (Mathematics Teacher, 1990 )

Let \(t\) be a tautology and \(p\) an arbitrary proposition. Give the truth value of each proposition. $$\sim t \wedge p$$

Prove by contradiction, where \(p\) is a prime number. \(\sqrt{p}\) is an irrational number.

Rewrite each sentence symbolically, where the UD consists of real numbers. The product of any two real numbers \(x\) and \(y\) is positive.

A family party consisted of one grandfather, one grandmother, two fathers, two mothers, four children, three grandchildren, one brother, two sisters, two sons, two daughters, one father-in-law, one mother-inlaw, and one daughter-in- law. A total of 23 people, apparently. But no; there were only seven people at the party. How could this be possible? (B. Hamilton, 1992 )

Let \(t\) be a tautology and \(p\) an arbitrary proposition. Give the truth value of each proposition. $$\sim(\sim p \wedge \sim t)$$

Prove by contradiction, where \(p\) is a prime number. \(\log _{10} 2\) is an irrational number.

Rewrite each sentence symbolically, where the UD consists of real numbers. There are real numbers \(x\) and \(y\) such that \(x=2 y\)

Prove by contradiction, where \(p\) is a prime number. \(\log _{10} 2\) is an irrational number.

Three gentlemen - Mr. Blue, Mr. Gray, and Mr. White-have shirts and ties that are blue, gray, and white, but not necessarily in that order. No person's clothing has the same color as his last name. Mr. Blue's tie has the same color as Mr. Gray's shirt. What color is Mr. White's shirt? (Mathematics Teacher, 1986)

Prove by cases, where \(n\) is an arbitrary integer and \(|x|\) denotes the absolute value of \(x\). \(n^{2}+n\) is an even integer.

Construct a truth table for each proposition. $$\sim p \vee \sim q$$

Rewrite each sentence symbolically, where the UD consists of real numbers. For each real number \(x,\) there is some real number \(y\) such that \(x \cdot y=x\)

Three men and their wives were given \(\$ 5400 .\) The wives together received \(\$ 2400 .\) Sue had \(\$ 200\) more than Jan, and Lynn had \(\$ 200\) more than Sue. Lou got half as much as his wife, Bob the same as his wife, and Matt twice as much as his wife. Who is married to whom? (Mathematics Teacher, 1986)

Prove by cases, where \(n\) is an arbitrary integer and \(|x|\) denotes the absolute value of \(x\). \(2 n^{3}+3 n^{2}+n\) is an even integer.

Construct a truth table for each proposition. $$\sim(\sim p \vee q)$$

Rewrite each sentence symbolically, where the UD consists of real numbers. There is a real number \(x\) such that \(x+y=y\) for every real number \(y\)

There are seven lots, 1 through \(7,\) to be developed in a certain city. A builder would like to build one bank, two hotels, and two restaurants on these lots, subject to the following restrictions by the city planning board (The Official LSAT PrepBook, 1991): If lot 2 is used, lot 4 cannot be used. If lot 5 is used, lot 6 cannot be used. The bank can be built only on lot \(5,6,\) or \(7 .\) A hotel cannot be built on lot \(5 .\) A restaurant can be built only on lot \(1,2,3,\) or \(5 .\) Which of the following is a possible list of locations for building them? A. The bank on lot \(7,\) hotels on lots 1 and \(4,\) and restaurants on lots 2 and 5 B. The bank on lot \(7,\) hotels on lots 3 and \(4,\) and restaurants on lots 1 and 5 C. The bank on lot \(7,\) hotels on lots 4 and \(5,\) and restaurants on lots 1 and 3.

Prove by cases, where \(n\) is an arbitrary integer and \(|x|\) denotes the absolute value of \(x\). \(n^{3}-n\) is divisible by \(3 .\) (Hint: Assume that every integer is of the form \(3 k, 3 k+1, \text { or } 3 k+2 .)\)

Construct a truth table for each proposition. $$(p \vee q) \vee(\sim q)$$

The logical operators NAND (not and) and NOR (not or) are defined as follows: $$ \begin{aligned} p & \text { NAND } q \equiv \sim(p \wedge q) \\ p & \text { NOR } q \equiv \sim(p \vee q) \end{aligned} $$ Construct a truth table for each proposition. \(p\) NAND \(q\)

There are seven lots, 1 through \(7,\) to be developed in a certain city. A builder would like to build one bank, two hotels, and two restaurants on these lots, subject to the following restrictions by the city planning board (The Official LSAT PrepBook, 1991): If lot 2 is used, lot 4 cannot be used. If lot 5 is used, lot 6 cannot be used. The bank can be built only on lot \(5,6,\) or \(7 .\) A hotel cannot be built on lot \(5 .\) A restaurant can be built only on lot \(1,2,3,\) or \(5 .\) If a restaurant is built on lot \(5,\) which of the following is not a possible list of locations? A. A hotel on lot 2 and lot 4 is left undeveloped. B. A restaurant on lot 2 and lot 4 is left undeveloped. C. A hotel on lot 2 and lot 3 is left undeveloped.

Prove by cases, where \(n\) is an arbitrary integer and \(|x|\) denotes the absolute value of \(x\). $$|-x|=|x|$$

Construct a truth table for each proposition. $$p \wedge(q \wedge r)$$

The logical operators NAND (not and) and NOR (not or) are defined as follows: $$ \begin{aligned} p & \text { NAND } q \equiv \sim(p \wedge q) \\ p & \text { NOR } q \equiv \sim(p \vee q) \end{aligned} $$ Construct a truth table for each proposition. \(p\) NOR \(q\)

Prove by cases, where \(n\) is an arbitrary integer and \(|x|\) denotes the absolute value of \(x\). $$|x \cdot y|=|x| \cdot|y|$$

Refer to Example 1.32 and are based on Smullyan's What is the name of this book? A and \(\mathrm{B}\) are inhabitants of the island. What are they if A says each of the following? "At least one of us is a knave."

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \wedge q \equiv q \wedge p$$

Prove by cases, where \(n\) is an arbitrary integer and \(|x|\) denotes the absolute value of \(x\). $$|x+y| \leq|x|+|y|$$

Give the truth value of each proposition, using the given information. \(p \wedge q,\) where \(\sim q\) is not false.

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \vee q \equiv q \vee p$$

Prove by the existence method. There are integers \(x\) such that \(x^{2}=x\)

Give the truth value of each proposition, using the given information. \(p \vee q,\) where \(\sim p\) is false.

Rewrite each in words, where UD = set of integers. $$(\forall x)\left(x^{2} \geq 0\right)$$

Prove by the existence method. There are integers \(x\) such that \(x^{2}=x\).

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \wedge t \equiv p$$

Prove by the existence method. There are integers \(x\) such that \(|x|=x\)

Give the truth value of each proposition, using the given information. \(p \vee q,\) where \(\sim p\) is not true.

Prove by the existence method. There are integers \(x\) such that \(|x|=x\).

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \vee f \equiv p$$

Refer to Example 1.32 and are based on Smullyan's What is the name of this book? A and \(\mathrm{B}\) are inhabitants of the island. What are they if A says each of the following? A says, "All of us are knaves," and B says, "Exactly one of us is a knave." What is C?

The exclusive disjunetion of two propositions \(p\) and \(q\) is denoted by \(p\) XOR \(q\) . Construct a truth table for \(p\) XOR \(q .\)

Rewrite each in words, where UD = set of integers. $$(\exists x)(\exists y)(x+y=7)$$