2026-09-15

Exercises for Section 2.1

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 2.1” (p. 38).

A. Decide whether or not the following are statements. In the case of a statement, say if it is true or false, if possible.

• 1. Every real number is an even integer.

• 4. Sets \(\mathbb{Z}\) and \(\mathbb{N}\).

• 11. The integer \(x\) is a multiple of 7.

Exercises for Section 2.2

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 2.2” (p. 42).

A. Express each statement or open sentence in a symbolic form such as \(P\wedge Q\), \(P\vee Q\), \(P\vee\sim Q\) or \(\sim P\), etc. Be sure to also state exactly what statements \(P\) and \(Q\) stand for.

• 1. The number 8 is both even and a power of 2.

• 6. There is a quiz scheduled for Wednesday or Friday.

• 9. \(x\in A-B\)

Exercises for Section 2.3

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 2.3” (pp. 45–46).

A. Without changing their meanings, convert each of the following sentences into a sentence having the form “If \(P\), then \(Q\).”

• 1. A matrix is invertible provided that its determinant is not zero.

• 5. An integer is divisible by 8 only if it is divisible by 4.

• 7. A series converges whenever it converges absolutely.