2026-09-10

Exercises for Section 1.3

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 1.3” (p. 15).

A. List all the subsets of the following sets.

• 1. \(\{1,2,3,4\}\)

Exercises for Section 1.4

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 1.4” (p. 17).

A. Write the following sets by listing their elements between braces.

• 2. \(\mathcal{P}(\{1,2,3,4\})\)

• 4. \(\mathcal{P}(\{\mathbb{R},\mathbb{Q}\})\)

• 7. \(\mathcal{P}(\{a,b\})\times\mathcal{P}(\{0,1\})\)

Exercises for Section 1.5

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 1.5” (pp. 19–20).

A. Suppose \(A\), \(B\), and \(C\) are the indicated sets. Find the following unions, intersections, and differences.

• 1. Suppose \(A=\{4,3,6,7,1,9\}\), \(B=\{5,6,8,4\}\), and \(C=\{5,8,4\}\).

  • (a) \(A\cup B\)

  • (d) \(A-C\)

  • (g) \(B\cap C\)

B. Sketch the following sets on the plane \(\mathbb{R}^2\). On separate drawings, shade in the union, intersection, and both differences.

• 6. \(X=[-1,3]\times[0,2]\) and \(Y=[0,3]\times[1,4]\)

C. Decide if the following statement is true or false. Justify your answer.

• 10. \((\mathbb{R}-\mathbb{Z})\times\mathbb{N}=(\mathbb{R}\times\mathbb{N})-(\mathbb{Z}\times\mathbb{N})\)