## Subsets and Power Sets

The following is the list of problems for Sections 1.3 and 1.4 of the Book of Proof (pages 14, 16). 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.

### Exercises for Section 1.3

A. List all the subsets of the following sets.

1. $$\{ 1, 2, 3, 4 \}$$
2. $$\{ 1, 2, \emptyset \}$$
3. $$\{ \{ \mathbb{R} \} \}$$
4. $$\emptyset$$
5. $$\{ \emptyset \}$$
6. $$\{ \mathbb{R}, \mathbb{Q}, \mathbb{N} \}$$
7. $$\{ \mathbb{R}, \{ \mathbb{Q}, \mathbb{N} \} \}$$
8. $$\{ \{ 0, 1 \}, \{ 0, 1, \{ 2 \} \}, \{ 0 \} \}$$

B. Write the following sets by listing their elements between braces.

1. $$\{ X : X \subseteq \{ 3, 2, a \} \text{ and } \lvert X \rvert = 2 \}$$
2. $$\{ X \subseteq \mathbb{N} : \lvert X \rvert \leq 1 \}$$
3. $$\{ X : X \subseteq \{ 3, 2, a \} \text{ and } \lvert X \rvert = 4 \}$$
4. $$\{ X : X \subseteq \{ 3, 2, a \} \text{ and } \lvert X \rvert = 1 \}$$

C. Decide if the following statements are true or false. Explain.

1. $$\mathbb{R}^3 \subseteq \mathbb{R}^3$$
2. $$\mathbb{R}^2 \subseteq \mathbb{R}^3$$
3. $$\{ (x,y) : x-1 =0 \} \subseteq \{ (x,y) : x^2-x=0 \}$$
4. $$\{ (x,y) : x^2-x =0 \} \subseteq \{ (x,y) : x-1=0 \}$$

### Exercises for Section 1.4

A. Find the indicated sets.

1. $$\mathscr{P}\big( \{ \{ a, b \}, \{ c \} \} \big)$$
2. $$\mathscr{P} \big( \{ 1, 2, 3, 4 \} \big)$$
3. $$\mathscr{P} \big( \{ \{ \emptyset \}, 5 \} \big)$$
4. $$\mathscr{P} \big( \{ \mathbb{R}, \mathbb{Q} \} \big)$$
5. $$\mathscr{P} \big(\mathscr{P} (\{ 2 \}) \big)$$
6. $$\mathscr{P} \big( \{ 1, 2 \} \big) \times\mathscr{P} (\{ 3 \})$$
7. $$\mathscr{P} \big( \{ a, b \} \big) \times\mathscr{P} \big(\{ 0, 1 \} \big)$$
8. $$\mathscr{P} \big( \{ 1, 2 \} \times \{ 3 \} \big)$$
9. $$\mathscr{P} \big( \{ a, b \} \times \{ 0 \} \big)$$
10. $$\{ X \in\mathscr{P} ( \{ 1, 2, 3 \} ) : \lvert X \rvert \leq 1 \}$$
11. $$\{ X \subseteq\mathscr{P} ( \{ 1, 2, 3 \} ) : \lvert X \rvert \leq 1 \}$$
12. $$\{ X \in\mathscr{P} ( \{ 1, 2, 3 \} ) : 2 \in X \}$$

B. Suppose that $$\lvert A \rvert = m$$ and $$\lvert B \rvert = n$$. Find the following cardinals.

1. $$\big\lvert\mathscr{P} \big(\mathscr{P} \big(\mathscr{P} (A) \big) \big) \big\rvert$$
2. $$\big\lvert\mathscr{P} \big(\mathscr{P} (A) \big) \big\rvert$$
3. $$\big\lvert\mathscr{P} (A \times B) \big\rvert$$
4. $$\big\lvert\mathscr{P}(A) \times\mathscr{P}(B) \big\rvert$$
5. $$\big\lvert \{ X \in\mathscr{P}(A) : \lvert X \rvert \leq 1 \} \big\rvert$$
6. $$\big\lvert\mathscr{P} \big( A \times\mathscr{P}(B) \big) \big\rvert$$
7. $$\big\lvert\mathscr{P} \big(\mathscr{P} \big(\mathscr{P} (A \times \emptyset) \big) \big) \big\rvert$$
8. $$\big\lvert \{ X \subseteq\mathscr{P}(A) : \lvert X \rvert \leq 1 \} \big\rvert$$