## 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.
1. 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. $$\displaystyle{\bigcap_{k=1}^3} A_k$$
2. 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. $$\displaystyle{\bigcap_{n \in \mathbb{N}}} A_n$$
3. For each $$n \in \mathbb{N}$$, let $$A_n = \{ -2n, 0, 2n \}$$.

1. $$\displaystyle{\bigcup_{n \in \mathbb{N}}} A_n$$
2. $$\displaystyle{\bigcap_{n \in \mathbb{N}}} A_n$$
1. $$\displaystyle{\bigcup_{n \in \mathbb{N}}} [0, n+1]$$
2. $$\displaystyle{\bigcap_{n \in \mathbb{N}}} [0, n+1]$$
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. $$\displaystyle{\bigcap_{\alpha \in \mathbb{R}}} \{ \alpha \} \times [0, 1]$$
1. $$\displaystyle{\bigcup_{x \in [0,1]}} [x,1] \times [0,x^2]$$
2. $$\displaystyle{\bigcap_{x \in [0,1]}} [x,1] \times [0,x^2]$$
1. 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 $$I$$?
2. 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 $$A_\alpha$$?
3. 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 $$\displaystyle{\bigcap_{\alpha \in J} A_\alpha \subseteq \bigcap_{\alpha \in I} A_\alpha}$$?
4. 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