2026-09-01

Exercises for Section 1.1

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Section 1.1” (pp. 7–8).

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

• 1. \(\{5x-1 : x\in\mathbb{Z}\}\)

• 11. \(\{x\in\mathbb{Z} : |x|<5\}\)

• 16. \(\{6a+2b : a,b\in\mathbb{Z}\}\)

B. Write each of the following sets in set-builder notation.

• 24. \(\{-4,-3,-2,-1,0,1,2\}\)

• 28. \(\left\{\ldots,-\frac{3}{2},-\frac{3}{4},0,\frac{3}{4},\frac{3}{2},\frac{9}{4},3,\frac{15}{4},\frac{9}{2},\ldots\right\}\)

C. Find the following cardinalities.

• 32. \(\left|\left\{\{\{1,4\},a,b,\{\{3,4\},\{\emptyset\}\}\}\right\}\right|\)

D. Sketch the following set of points in the \(x\)-\(y\) plane.

• 41. \(\{(x,y) : x\in[-1,1], y=1\}\)