2026-09-03

Exercises for Section 1.2

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 1.2” (p. 11).

A. Write out the indicated sets by listing their elements between braces.

• 1. Suppose \(A=\{1,2,3,4\}\) and \(B=\{a,c\}\).

  • (a) \(A\times B\)

  • (e) \(\emptyset\times B\)

• 2. Suppose \(A=\{\pi,e,0\}\) and \(B=\{0,1\}\).

  • (e) \(A\times \emptyset\)

  • (g) \(A\times(B\times B)\)

• 4. \(\{n\in\mathbb{Z} : 2<n<5\}\times\{n\in\mathbb{Z} : |n|=5\}\)

• 7. \(\{\emptyset\}\times\{0,\emptyset\}\times\{0,1\}\)

B. Sketch these Cartesian products on the \(x\)-\(y\) plane \(\mathbb{R}^2\) (or \(\mathbb{R}^3\) for the last two).

• 12. \([-1,1]\times[1,2]\)

• 18. \(\mathbb{Z}\times\mathbb{Z}\)