MATH 300 Fall 2018 Assignment 03

Aug 14, 2018

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. \( \big\lvert \{ X \subseteq\mathscr{P}(A) : \lvert X \rvert \leq 1 \} \big\rvert \)