Book of Proof - Amazon S3

Solutions
Chapter 1 Exercises
Section 1.1
1. {5x − 1 : x ∈ Z} = {... − 11,−6,−1,4,9,14,19,24,29,...}
3. {x ∈ Z : −2 ≤ x < 7} = {−2,−1,0,1,2,3,4,5,6}
5. { x ∈ R : x 2 = 3 } = { − 3, 3 }
7. { x ∈ R : x 2 + 5x = −6 } = {−2,−3}
9. {x ∈ R : sinπx = 0} = {...,−2,−1,0,1,2,3,4,...} = Z
11. {x ∈ Z : |x| < 5} = {−4,−3,−2,−1,0,1,2,3,4}
13. {x ∈ Z : |6x| < 5} = {0}
15. {5a + 2b : a, b ∈ Z} = {...,−2,−1,0,1,2,3,...} = Z
17. {2,4,8,16,32,64...} = {2 x : x ∈ N}
19. {...,−6,−3,0,3,6,9,12,15,...} = {3x : x ∈ Z}
21. {0,1,4,9,16,25,36,...} = { x 2 : x ∈ Z }
23. {3,4,5,6,7,8} = {x ∈ Z : 3 ≤ x ≤ 8} = {x ∈ N : 3 ≤ x ≤ 8}
25. { ..., 1 8 , 1 4 , 1 2 ,1,2,4,8,...} = {2 n : n ∈ Z}
27. { { }
...,−π,− π 2 ,0, π 2 ,π, 3π 2 ,2π, 5π 2 ,...} = kπ
2 : k ∈ Z
29. |{{1},{2,{3,4}},}| = 3
31. |{{{1},{2,{3,4}},}}| = 1
39. {(x, y) : x ∈ [1,2], y ∈ [1,2]}
33. |{x ∈ Z : |x| < 10}| = 19
35. | { x ∈ Z : x 2 < 10 } | = 7
37. | { x ∈ N : x 2 < 0 } | = 0
43. {(x, y) : |x| = 2, y ∈ [0,1]}
2
1
−3 −2 −1 1 2 3
−1
−2
41. {(x, y) : x ∈ [−1,1], y = 1}
2
1
−3 −2 −1 1 2 3
−1
−2
2
1
−3 −2 −1 1 2 3
−1
−2
45. { (x, y) : x, y ∈ R, x 2 + y 2 = 1 }
2
1
−3 −2 −1 1 2 3
−1
−2