Statements

The following is the list of problems for Section 1.8 of the Book of Proof (page 28). 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.

2. Suppose \( A_1 = \{ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 \} \), \( A_2 = \{ 0, 3, 6, 9, 12, 15, 18, 21, 24 \} \), \( A_3 = \{ 0, 4, 8, 12, 16, 20, 24 \} \).

   1. \( \displaystyle{\bigcup_{k=1}^3 A_k} \)
   2. \( \displaystyle{\bigcap_{k=1}^3} A_k \)

3. For each \( n \in \mathbb{N} \), let \( A_n = \{ 0, 1, 2, 3, \dotsc, n \} \).

   1. \( \displaystyle{\bigcup_{n \in \mathbb{N}}} A_n \)
   2. \( \displaystyle{\bigcap_{n \in \mathbb{N}}} A_n \)

4. For each \( n \in \mathbb{N} \), let \( A_n = \{ -2n, 0, 2n \} \).

   1. \( \displaystyle{\bigcup_{n \in \mathbb{N}}} A_n \)
   2. \( \displaystyle{\bigcap_{n \in \mathbb{N}}} A_n \)

6. 1. \( \displaystyle{\bigcup_{n \in \mathbb{N}}} [0, n+1] \)
   2. \( \displaystyle{\bigcap_{n \in \mathbb{N}}} [0, n+1] \)
7. 1. \( \displaystyle{\bigcup_{n \in \mathbb{N}}} \mathbb{R} \times [n, n+1] \)
   2. \( \displaystyle{\bigcap_{n \in \mathbb{N}}} \mathbb{R} \times [n, n+1] \)

   8.

   1. \( \displaystyle{\bigcup_{\alpha \in \mathbb{R}}} \{ \alpha \} \times [0, 1] \)
   2. \( \displaystyle{\bigcap_{\alpha \in \mathbb{R}}} \{ \alpha \} \times [0, 1] \)

10. 1. \( \displaystyle{\bigcup_{x \in [0,1]}} [x,1] \times [0,x^2] \)
    2. \( \displaystyle{\bigcap_{x \in [0,1]}} [x,1] \times [0,x^2] \)
11. Is \( \displaystyle{\bigcap_{\alpha \in I} A_\alpha \subseteq \bigcup_{\alpha \in I} A_\alpha} \) always true for any collection of sets \( A_\alpha \) with index set \( I \)?
12. If \( \displaystyle{\bigcap_{\alpha \in I} A_\alpha = \bigcup_{\alpha \in I} A_\alpha} \), what do you think can be said about the relationships between the sets \( A_\alpha \)?
13. If \( J \neq \emptyset \) and \( J \subseteq I \), does it follow that \( \displaystyle{ \bigcup_{\alpha \in J} A_\alpha \subseteq \bigcup_{\alpha \in I} A_\alpha} \)? What about \( \displaystyle{\bigcap_{\alpha \in J} A_\alpha \subseteq \bigcap_{\alpha \in I} A_\alpha} \)?
14. If \( J \neq \emptyset \) and \( J \subseteq I \), does it follow that \( \displaystyle{\bigcap_{\alpha \in I} A_\alpha \subseteq \bigcap_{\alpha \in J} A_\alpha} \)? Explain