Introduction to Sets

The following is the list of problems for Section 1.1 of the Book of Proof (page 7). There is a forum open at the end, so you can ask questions. It is a great way to interact with the instructor and with other students in your class, should you need some assistance with any question. Please, do not post solutions.

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

1. \( \{ 5x-1 : x \in \mathbb{Z} \} \)
2. \( \{ 3x+2 : x \in \mathbb{Z} \} \)
3. \( \{ x \in \mathbb{Z} : -2 \leq x < 7\} \)
4. \( \{ x \in \mathbb{N} : -2 < x \leq 7 \} \)
5. \( \{ x \in \mathbb{R} : x^2=3 \} \)
6. \( \{ x \in \mathbb{R} : x^2=9 \} \)
7. \( \{ x \in \mathbb{R} : x^2+5x=-6 \} \)
8. \( \{ x \in \mathbb{R} : x^3+5x^2=-6x \} \)
9. \( \{ x \in \mathbb{R} : \sin(\pi x) = 0 \} \)
10. \( \{ x \in \mathbb{R} : \cos(x) = 1 \} \)
11. \( \{ x \in \mathbb{Z} : \lvert x \rvert < 5 \} \)
12. \( \{ x \in \mathbb{Z} : \lvert 2x \rvert < 5 \} \)
13. \( \{ x \in \mathbb{Z} : \lvert 6x \rvert < 5 \} \)
14. \( \{ 5x : x \in \mathbb{Z}, \lvert 2x \rvert \leq 8 \} \)
15. \( \{ 5a + 2b : a,b \in \mathbb{Z} \} \)
16. \( \{ 6a + 2b : a,b \in \mathbb{Z} \} \)

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

17. \( \{ 2, 4, 8, 16, 32, 64, \dotsc \} \)
18. \( \{ 0, 4, 16, 36, 64, 100, \dotsc \} \)
19. \( \{ \dotsc, -6, -3, 0 , 3, 6, 9, 12, 15, \dotsc \} \)
20. \( \{ \dotsc, -8, -3, 2 , 7, 12, 17, \dotsc \} \)
21. \( \{ 0, 1, 4, 9, 16, 25, 36, \dotsc \} \)
22. \( \{ 3, 6, 11, 18, 27, 38, \dotsc \} \)
23. \( \{ 3, 4, 5, 6, 7, 8 \} \)
24. \( \{ -4, -3, -2, -1, 0, 1, 2 \} \)
25. \( \{ \dotsc, \tfrac{1}{8}, \tfrac{1}{4}, \tfrac{1}{2}, 1, 2, 4, 8, \dotsc \} \)
26. \( \{ \dotsc, \tfrac{1}{27}, \tfrac{1}{9}, \tfrac{1}{3}, 1, 3, 9, 27 \} \)
27. \( \{ \dotsc, -\pi, -\tfrac{\pi}{2}, 0, \tfrac{\pi}{2}, \pi, \tfrac{3\pi}{2}, 2\pi, \tfrac{5\pi}{2} \} \)
28. \( \{ \dotsc, -\tfrac{3}{2}, -\tfrac{3}{4}, 0, \tfrac{3}{4}, \tfrac{3}{2}, \tfrac{9}{4}, 3, \tfrac{15}{4}, \tfrac{9}{2} \} \)

C. Find the following cardinalities.

29. \( \big\lvert \{ \{ 1 \}, \{ 2, \{ 3, 4 \} \}, \emptyset \} \big\rvert \)
30. \( \big\lvert \{ \{ 1, 4 \}, a, b, \{ \{ 3, 4 \} \}, \{ \emptyset \} \} \big\rvert \)
31. \( \big\lvert \{ \{ \{ 1 \}, \{ 2, \{ 3, 4 \} \}, \emptyset \} \} \big\rvert \)
32. \( \big\lvert \{ \{ \{ 1, 4 \}, a, b, \{ \{ 3, 4 \} \}, \{ \emptyset \} \} \} \big\rvert \)
33. \( \big\lvert \{ x \in \mathbb{Z} : \lvert x \rvert < 10 \} \big\rvert \)
34. \( \big\lvert \{ x \in \mathbb{N} : \lvert x \rvert < 10 \} \big\rvert \)
35. \( \big\lvert \{ x \in \mathbb{Z} : x^2 < 10 \} \big\rvert \)
36. \( \big\lvert \{ x \in \mathbb{N} : x^2 < 10 \} \big\rvert \)
37. \( \big\lvert \{ x \in \mathbb{N} : x^2 < 0 \} \big\rvert \)
38. \( \big\lvert \{ x \in \mathbb{N} : 5x \leq 20 \} \big\rvert \)

D. Sketch the following sets of points in the \( xy \)—plane.

39. \( \{ (x,y) : x \in [1,2], y \in [1,2] \} \)
40. \( \{ (x,y) : x \in [0,1], y \in [1,2] \} \)
41. \( \{ (x,y) : x \in [-1,1], y=1 \} \)
42. \( \{ (x,y) : x=2, y \in [0,1] \} \)
43. \( \{ (x,y) : \lvert x \rvert = 2, y \in [0,1] \} \)
44. \( \{ (x,x^2) : x \in \mathbb{R} \} \)
45. \( \{ (x,y) : x,y \in \mathbb{R}, x^2+y^2=1 \} \)
46. \( \{ (x,y) : x,y \in \mathbb{R}, x^2+y^2\leq 1 \} \)
47. \( \{ (x,y) : x,y \in \mathbb{R}, y \geq x^2 - 1 \} \)
48. \( \{ (x,y) : x,y \in \mathbb{R}, x > 1 \} \)
49. \( \{ (x, x+y) : x \in \mathbb{R}, y \in \mathbb{Z} \} \)
50. \( \big\{ \big(x, \tfrac{x^2}{y} \big) : x \in \mathbb{R}, y \in \mathbb{N} \big\} \)
51. \( \{ (x,y) \in \mathbb{R}^2 : (y-x)(y+x)=0 \} \)
52. \( \{ (x,y) \in \mathbb{R}^2 : (y-x^2)(y+x^2)=0 \} \)