The Cartesian Product

The following is the list of problems for Section 1.2 of the Book of Proof (page 10). 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 out the indicated sets by listing their elements between braces.

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

   1. \( A \times B \)
   2. \( B \times A \)
   3. \( A \times A \)
   4. \( B \times B \)
   5. \( \emptyset \times B \)
   6. \( (A \times B) \times B \)
   7. \( A \times (B \times B) \)
   8. \( B^3 \)

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

   1. \( A \times B \)
   2. \( B \times A \)
   3. \( A \times A \)
   4. \( B \times B \)
   5. \( \emptyset \times B \)
   6. \( (A \times B) \times B \)
   7. \( A \times (B \times B) \)
   8. \( A \times B \times B \)
3. \( \{ x \in \mathbb{R} : x^2 = 2 \} \times \{ a, c, e \} \)
4. \( \{ n \in \mathbb{Z} : 2 < n < 5 \} \times \{ n \in \mathbb{Z} : \lvert n \rvert = 5 \} \)
5. \( \{ x \in \mathbb{R} : x^2 = 2 \} \times \{ x \in \mathbb{R} : \lvert x \rvert = 2 \} \)
6. \( \{ x \in \mathbb{R} : x^2=x \} \times \{ x \in \mathbb{N} : x^2=x \} \)
7. \( \{ \emptyset \} \times \{ 0, \emptyset \} \times \{ 0, 1 \} \)
8. \( \{ 0, 1 \}^4 \)

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

9. \( \{ 1, 2, 3 \} \times \{ -1, 0 , 1\} \)
10. \( \{ -1, 0, 1 \} \times \{ 1, 2, 3 \} \)
11. \( [0,1] \times [0,1] \)
12. \( [-1, 1] \times [1,2] \)
13. \( \{ 1, 1.5, 2 \} \times [1,2] \)
14. \( [1,2] \times \{ 1, 1.5, 2 \} \)
15. \( \{ 1 \} \times [0,1] \)
16. \( [0,1] \times \{ 1 \} \)
17. \( \mathbb{N} \times \mathbb{Z} \)
18. \( \mathbb{Z} \times \mathbb{Z} \)
19. \( [0,1] \times [0,1] \times [0,1] \)
20. \( \{ (x,y) \in \mathbb{R}^2 : x^2+y^2\leq 1 \} \times [0,1] \)