2026-09-17
Exercises for Section 2.4
Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 2.4” (p. 47).
A. Without changing their meanings, convert each of the following sentences into a sentence having the form “P if and only if Q.”
1. For matrix \(A\) to be invertible, it is necessary and sufficient that \(\det(A)\neq 0\).
2. If a function has a constant derivative then it is linear, and conversely.
3. If \(xy=0\) then \(x=0\) or \(y=0\), and conversely.
4. If \(a\in\mathbb{Q}\) then \(5a\in\mathbb{Q}\), and if \(5a\in\mathbb{Q}\) then \(a\in\mathbb{Q}\).
Exercises for Section 2.5
Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 2.5” (p. 50).
A. Write a truth table for each of the following logical statements.
2. \((Q\vee R)\Leftrightarrow(R\wedge Q)\)
4. \(\sim(P\vee Q)\vee(\sim P)\)
8. \(P\vee(Q\wedge\sim R)\)
B. Find the truth values of the letters below. (This can be done without a truth table.)
- 10. Suppose the statement \(((P\wedge Q)\vee R)\Rightarrow(R\vee S)\) is false. Find the truth values of \(P\), \(Q\), \(R\), and \(S\).
Exercises for Section 2.6
Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 2.6” (p. 52).
A. Use truth tables to show that the following statements are logically equivalent.
3. \(P\Rightarrow Q = (\sim P)\vee Q\)
4. \(\sim(P\vee Q) = (\sim P)\wedge(\sim Q)\)
B. Decide whether or not the following pairs of statements are logically equivalent.
12. \(\sim(P\Rightarrow Q)\) and \(P\wedge\sim Q\)
13. \(P\vee(Q\wedge R)\) and \((P\vee Q)\wedge R\)