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**Exercise Set 6.1**

1. In each of (a) -(f), answer the following questions: Is $A \subseteq B$? Is
   $B \subseteq A$? Is either $A$ or $B$ a proper subset of the other?

a. $A = \{2, \{2\}, (\sqrt{2})^2\}$, $B = \{2, \{2\}, \{\{2\}\}\}$

b. $A = \{3, \sqrt{5^2 - 4^2}, 24 \mod 7\}$, $B = \{8 \mod 5\}$

c. $A = \{\{1, 2\}, \{2, 3\}\}$, $B = \{1, 2, 3\}$

d. $A = \{a, b, c\}$, $B = \{\{a\}, \{b\}, \{c\}\}$

e. $A = \{\sqrt{16}, \{4\}\}$, $B = \{4\}$

f. $A = \{x \in \mathbb{R} | \cos x \in \mathbb{Z}\}$,
$B = \{x \in \mathbb{R} | \sin x \in \mathbb{Z}\}$

2. Complete the proof from Example 6.1.3: Prove that $B \subseteq A$ where

$$ A = \{m \in \mathbb{Z} | m, = 2a \text{ for some integer } a\} $$

and

$$ B = \{n \in \mathbb{Z} | n = 2b - 2 \text{ for some integer } b\} $$

3. Let sets $R$, $S$, and $T$ be defined as follows:

$$ R = \{x \in \mathbb{Z} | x \text{ is divisible by } 2\}  $$

$$ S = \{y \in \mathbb{Z} | y \text{ is divisible by } 3\}  $$

$$ T = \{z \in \mathbb{Z} | z \text{ is divisible by } 6\}  $$

Prove or disprove each of the following statements.

a. $R \subseteq T$

b. $T \subseteq R$

c. $T \subseteq S$

4. Let $A = \{n \in \mathbb{Z} | n = 5r \text{ for some integer } r\}$ and
   $B = \{m \in \mathbb{Z} | m = 20s \text{ for some integer } s\}$. Prove or
   disprove each of the following statements.

a. $A \subseteq B$

b. $B \subseteq A$

5. Let $C = \{n \in \mathbb{Z} | n = 6r - 5 \text{ for some integer } r\}$ and
   $D = \{m \in \mathbb{Z} | m = 3s + 1 \text{ for some integer } s\}$. Prove or
   disprove each of the following statements.

a. $C \subseteq D$

b. $D \subseteq C$

6. Let $A = \{x \in \mathbb{Z} | x = 5a + 2 \text{ for some integer } a\}$,
   $B = \{y \in \mathbb{Z} | y = 10b - 3 \text{ for some integer } b\}$, and
   $C = \{z \in \mathbb{Z} | z = 10c + 7 \text{ for some integer } c\}$.

Prove or disprove each of the following statements.

a. $A \subseteq B$

b. $B \subseteq A$

c. $B = C$

7. Let $A = \{x \in \mathbb{Z} | x = 6a + 4 \text{ for some integer } a\}$,
   $B = \{y \in \mathbb{Z} | y = 18b - 2 \text{ for some integer } b\}$, and
   $C = \{z \in \mathbb{Z} | z = 18c + 16 \text{ for some integer } c\}$.

Prove or disprove each of the following statements.

a. $A \subseteq B$

b. $B \subseteq A$

c. $B = C$

8. Write in words to read each of the following out loud. Then write each set
   using the symbols for union, intersection, set difference, or set complement.

a. $\{x \in U | x \in A \text{ and } x \in B\}$

b. $\{x \in U | x \in A \text{ or } x \in B\}$

c. $\{x \in U | x \in A \text{ and } x \notin B\}$

d. $\{x \in U | x \notin A\}$

9. Complete the following sentences without using the symbols $\cup$, $\cap$, or
   $-$.

a. $x \notin A \cup B$ if, and only if, _____.

b. $x \notin A \cap B$ if, and only if, _____.

c. $x \notin A - B$ if, and only if, _____.

10. Let $A = \{1, 3, 5, 7, 9\}$, $b = \{3, 6, 9\}$, and $C = \{2, 4, 6, 8\}$.
    Find each of the following:

a. $A \cup B$

b. $A \cap B$

c. $A \cup C$

d. $A \cap C$

e. $A - B$

f. $B - A$

g. $B \cup C$

h. $B \cap C$

11. Let the universal set $\mathbb{R}$, the set of all real numbers, and let
    $A = \{x \in \mathbb{R} | 0 < x \leq 2\}$,
    $B = \{x \in \mathbb{R} | 1 \leq x < 4\}$, and
    $C = \{x \in \mathbb{R} | 3 \leq x < 9\}$. Find each of the following:

a. $A \cup B$

b. $A \cap B$

c. $A^c$

d. $A \cup C$

e. $A \cap C$

f. $B^c$

g. $A^c \cap B^c$

h. $A^c \cup B^c$

i. $(A \cap B)^c$

j. $(A \cup B)^c$

12. Let the universal set be $\mathbb{R}$, the set of all real numbers, and let
    $A = \{x \in \mathbb{R} | -3 \leq x \leq 0\}$,
    $B = \{x \in \mathbb{R} | -1 < x < 2\}$, and
    $C = \{x \in \mathbb{R} | 6 < x \leq 8\}$. Find each of the following:

a. $A \cup B$

b. $A \cap B$

c. $A^c$

d. $A \cup C$

e. $A \cap C$

f. $B^c$

g. $A^c \cap B^c$

h. $A^c \cup B^c$

i. $(A \cap B)^c$

j. $(A \cup B)^c$

13. Let $S$ be the set of all strings of $0$'s and $1$'s of length $4$, and let
    $A$ and $B$ be the following subsets of $S$:
    $A = \{1110, 1111, 1000, 1001\}$ and $B = \{1100, 0100, 1111, 0111\}$. Find
    each of the following:

a. $A \cap B$

b. $A \cup B$

c. $A - B$

d. $B - A$

14. In each of the following, draw a Venn diagram for sets $A$, $B$, and $C$
    that satisfy the given conditions.

a. $A \subseteq B$, $C \subseteq B$, $A \cap C = \emptyset$

b. $C \subseteq A$, $B \cap C = \emptyset$

15. In each of the following, draw a Venn diagram for sets $A$, $B$, and $C$
    that satisfy the given conditions.

a. $A \cap B = \emptyset$, $A \subseteq C$, $C \cap B \neq \emptyset$

b. $A \subseteq B$, $C \subseteq B$, $A \cap C \neq \emptyset$

c. $A \cap B \neq \emptyset$, $B \cap C \neq \emptyset$, $A \cap C = \emptyset$,
$A \nsubseteq B$, $C \nsubseteq B$

16. Let $A = \{a, b, c\}$, $B = \{b, c, d\}$, and $C = \{b, c, e\}$.

a. Find $A \cup (B \cap C)$, $(A \cup B) \cap C$, and
$(A \cup B) \cap (A \cup C)$. Which of these sets are equal?

b. Find $A \cap (B \cup C)$, $(A \cap B) \cup C$, and
$(A \cap B) \cup (A \cap C)$. Which of these sets are equal?

c. Find $(A - B) - C$ and $A - (B - C)$. Are these sets equal?

17. Consider the following Venn diagram. For each of (a)-(f), copy the diagram
    and shade the region corresponding to the indicated set.

a. $A \cap B$

b. $B \cup C$

c. $A^c$

d. $A - (B \cup C)$

e. $(A \cup B)^c$

f. $A^c \cap B^c$

(See page 412 for image)

18.

a. Is the number $0$ in $\emptyset$? Why?

b. Is $\emptyset = \{\emptyset\}$? Why?

c. Is $\emptyset \in \{\emptyset\}$ Why?

d. Is $\emptyset \in \emptyset$? Why?

19. Let $A_i = \{i, i^2\}$ for each integer $i = 1, 2, 3, 4$.

a. $A_1 \cup A_2 \cup A_3 \cup A_4 = \text{ ?}$

b. $A_1 \cap A_2 \cap A_3 \cap A_4 = \text{ ?}$

c. Are $A_1, A_2, A_3$, and $A_4$ mutually disjoint? Explain.

20. Let $B_i = \{x \in \mathbb{R} | 0 \leq x\leq i\}$ for each integer
    $i = 1, 2, 3, 4$.

a. $B_1 \cup B_2 \cup B_3 \cup B_4 = \text{ ?}$

b. $B_1 \cap B_2 \cap B_3 \cap B_4 = \text{ ?}$

c. Are $B_1, B_2, B_3$, and $B_4$ mutually disjoint? Explain.

21. Let $C_i = \{i, -i\}$ for each nonnegative integer $i$.

a. $\bigcup_{i = 0}^{4}C_i = \text{ ?}$

b. $\bigcap_{i = 0}^{4}C_i = \text{ ?}$

c. Are $C_0, C_1, C_2, \dots$ mutually disjoint? Explain.

d. $\bigcup_{i = 0}^{n}C_i = \text{ ?}$

e. $\bigcap_{i = 0}^{n}C_i = \text{ ?}$

f. $\bigcup_{i = 0}^{\infty}C_i = \text{ ?}$

g. $\bigcap_{i = 0}^{\infty}C_i = \text{ ?}$

22. Let $D_i = \{x \in \mathbb{R} | -i \leq x \leq i\} = [-i, i]$ for each
    nonnegative integer $i$.

a. $\bigcup_{i = 0}^{4}D_i = \text{ ?}$

b. $\bigcap_{i = 0}^{4}D_i = \text{ ?}$

c. Are $D_0, D_1, D_2, \dots$ mutually disjoint? Explain.

d. $\bigcup_{i = 0}^{n}D_i = \text{ ?}$

e. $\bigcap_{i = 0}^{n}D_i = \text{ ?}$

f. $\bigcup_{i = 0}^{\infty}D_i = \text{ ?}$

g. $\bigcap_{i = 0}^{\infty}D_i = \text{ ?}$

23. Let
    $V_i = \{x \in \mathbb{R} | -\dfrac{1}{i} \leq x \leq \dfrac{1}{i}\} = \left[-\dfrac{1}{i}, \dfrac{1}{i}\right]$
    for each positive integer $i$.

a. $\bigcup_{i = 0}^{4}V_i = \text{ ?}$

b. $\bigcap_{i = 0}^{4}V_i = \text{ ?}$

c. Are $V_1, V_2, V_3, \dots$ mutually disjoint? Explain.

d. $\bigcup_{i = 0}^{n}V_i = \text{ ?}$

e. $\bigcap_{i = 0}^{n}V_i = \text{ ?}$

f. $\bigcup_{i = 0}^{\infty} = \text{ ?}$

g. $\bigcap_{i = 0}^{\infty} = \text{ ?}$

24. Let $W_i = \{x \in \mathbb{R} | x > i\} = (i, \infty)$ for each nonnegative
    integer $i$.

a. $\bigcup_{i = 0}^{4}W_i = \text{ ?}$

b. $\bigcap_{i = 0}^{4}W_i = \text{ ?}$

c. Are $W_0, W_1, W_2, \dots$ mutually disjoint? Explain.

d. $\bigcup_{i = 0}^{n}W_i = \text{ ?}$

e. $\bigcap_{i = 0}^{n}W_i = \text{ ?}$

f. $\bigcup_{i = 0}^{\infty}W_i = \text{ ?}$

g. $\bigcap_{i = 0}^{\infty}W_i = \text{ ?}$

25. Let
    $R_i = \{x \in \mathbb{R} | 1 \leq x \leq 1 + \dfrac{1}{i}\} = \left[1, 1 + \dfrac{1}{i}\right]$
    for each positive integer $i$.

a. $\bigcup_{i = 0}^{4}R_i = \text{ ?}$

b. $\bigcap_{i = 0}^{4}R_i = \text{ ?}$

c. Are $R_1, R_2, R_3, \dots$ mutually disjoint? Explain.

d. $\bigcup_{i = 0}^{n}R_i = \text{ ?}$

e. $\bigcap_{i = 0}^{n}R_i = \text{ ?}$

f. $\bigcup_{i = 0}^{\infty}R_i = \text{ ?}$

g. $\bigcap_{i = 0}^{\infty}R_i = \text{ ?}$

26. Let
    $S_i = \{x \in \mathbb{R} | 1 < x < 1 + \dfrac{1}{i}\} = \left(1, 1 + \dfrac{1}{i}\right)$
    for each positive integer $i$.

a. $\bigcup_{i = 0}^{4}S_i = \text{ ?}$

b. $\bigcap_{i = 0}^{4}S_i = \text{ ?}$

c. Are $S_1, S_2, S_3, \dots$ mutually disjoint? Explain.

d. $\bigcup_{i = 0}^{n}S_i = \text{ ?}$

e. $\bigcap_{i = 0}^{n}S_i = \text{ ?}$

f. $\bigcup_{i = 0}^{\infty}S_i = \text{ ?}$

g. $\bigcap_{i = 0}^{\infty}S_i = \text{ ?}$

27.

a. Is $\{\{a, d, e\}, \{b, c\}, \{d, f\}\}$ a partition of
$\{a, b, c, d, e, f\}$?

b. Is $\{\{w, x, v\}, \{u, y, q\}, \{p, z\}\}$ a partition of
$\{p, q, u, v, w, x, y, z\}$?

c. Is $\{\{5, 4\}, \{7, 2\}, \{1, 3, 4\}, \{6, 8\}\}$ a partition of
$\{1, 2, 3, 4, 5, 6, 7, 8\}$?

d. Is $\{\{3, 7, 8\}, \{2, 9\}, \{1, 4, 5\}\}$ a partition of
$\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$?

e. Is $\{\{1, 5\}, \{4, 7\}, \{2, 8, 6, 3\}\}$ a partition of
$\{1, 2, 3, 4, 5, 6, 7, 8\}$?

28. Let $E$ be the set of all even integers and $O$ the set of all odd integers.
    Is $\{E, O\}$ a partition of $\mathbb{Z}$, the set of all integers? Explain
    your answer.

29. Let $\mathbb{R}$ be the set of all real numbers. Is
    $\{\mathbb{R}^+, \mathbb{R}^-, \{0\}\}$ a partition of $\mathbb{R}$? Explain
    your answer.

30. Let $\mathbb{Z}$ be the set of all integers and let

$$ A_0 = \{n \in \mathbb{Z} | n = 4k, \text{ for some integer } k\} $$

$$ A_1 = \{n \in \mathbb{Z} | n = 4k + 1, \text{ for some integer } k\} $$

$$ A_2 = \{n \in \mathbb{Z} | n = 4k + 2, \text{ for some integer } k\} $$

and

$$ A_3 = \{n \in \mathbb{Z} | n = 4k + 3, \text{ for some integer } k\} $$

Is $\{A_0, A_1, A_2, A_3\}$ a partition of $\mathbb{Z}$? Explain your answer.

31. Suppose $A = \{1, 2\}$ and $B = \{2, 3\}$. Find each of the following:

a. $\mathscr{P}(A \cap B)$

b. $\mathscr{P}(A)$

c. $\mathscr{P}(A \cup B)$

d. $\mathscr{P}(A \times B)$

32.

a. Suppose $A = \{1\}$ and $B = \{u, v\}$. Find $\mathscr{P}(A \times B)$.

b. Suppose $X = \{a, b\}$ and $Y = \{x, y\}$. Find $\mathscr{P}(X \times Y)$.

33.

a. Find $\mathscr{P}(\emptyset)$.

b. Find $\mathscr{P}(\mathscr{P}(\emptyset))$.

b. Find $\mathscr{P}(\mathscr{P}(\mathscr{P}(\emptyset)))$.

34. let $A_1 = \{1\}$, $A_2 = \{u, v\}$, and $A_3 = \{m, n\}$. Find each of the
    following sets:

a. $A_1 \cup (A_2 \times A_3)$

b. $(A_1 \cup A_2) \times A_3$

35. let $A = \{a, b\}$, $B = \{1, 2\}$, and $C = \{2, 3\}$. Find each of the
    following sets:

a. $A \times (B \cup C)$

b. $(A \times B) \cup (A \times C)$

c. $A \times (B \cap C)$

d. $(A \times B) \cap (A \times C)$

36. Trace the action of Algorithm 6.1.1 on the variables $i$, $j$,
    $\text{found}$, and $\text{answer}$ for $m = 3$, $n = 3$, and sets $A$ and
    $B$ represented as the arrays
    $a[1] = u, a[2] = v, a[3] = w, b[1] = w, b[2] = u,$ and $b[3] = v$.

37. Trace the action of Algorithm 6.1.1 on the variables $i$, $j$,
    $\text{found}$, and $\text{answer}$ for $m = 4$, $n = 4$ and sets $A$ and
    $B$ represented as the arrays
    $a[1] = u, a[2] = v, a[3] = w, a[4] = x, b[1] = r, b[2] = u, b[3] = y, b[4] = z$.

38. Write an algorithm to determine whether a given element $x$ belongs to a
    given set that is represented as the array $a[1], a[2], \dots, a[n]$.