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Page 411
**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 $a[1], a[2], \dots, a[n]$.

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Page 401
**Element Argument: The Basic Method for Proving That One Set is a Subset of
Another**
Let sets $X$ and $Y$ be given. To prove that $X \subseteq Y$,
1. **suppose** that $x$ is a particular but arbitrarily chosen element of $X$,
2. **show** that $x$ is an element of $Y$.
---
Page 402
**Definition**
Given sets $A$ and $B$, $A$ **equals** $B$, written $A = B$, if, and only if,
every element of $A$ is in $B$ and every element of $B$ is in $A$.
Symbolically:
$$ A = B \Leftrightarrow A \subseteq B \text{ and } B \subseteq A $$
---
Page 404
Let $A$ and $B$ be the subsets of a universal set $U$.
1. The **union** of $A$ and $B$, denoted $A \cup B$, is the set of all elements
that are in at least one of $A$ or $B$.
2. The **intersection of $A$ and $B$, denoted $A \cap B$, is the set of all
elements that are common to both $A$ and $B$.
3. The **difference** of $B$ minus $A$ (or **relative complement** of $A$ in
$B$), denoted $B - A$, is the set of all elements that are in $B$ and not
$A$.
4. The **complement** of $A$, denoted $A^c$, is the set of all elements in $U$
that are not in $A$.
Symbolically:
$$ A \cup B = \{x \in U | x \in A \text{ or } x \in B\} $$
$$ A \cap B = \{x \in U | x \in A \text{ and } x \in B\} $$
$$ B - A = \{x \in U | x \in B \text{ and } x \notin A\} $$
$$ A^c = \{x \in U | x \notin A\} $$
---
Page 405:
**Interval Notation:**
Given real numbers $a$ and $b$ with $a \leq b$:
$$ (a, b) = \{x \in \mathbb{R} | a < x < b\} $$
$$ [a, b] = \{x \in \mathbb{R} | a \leq x \leq b\} $$
$$ (a, b] = \{x \in \mathbb{R} | a < x \leq b\} $$
$$ [a, b) = \{x \in \mathbb{R} | a \leq x < b\} $$
The symbols $\infty$ and $-\infty$ are used to indicate intervals that are
unbounded either on the right or on the left:
$$ (a, \infty) = \{x \in \mathbb{R} | x > a\} $$
$$ [a, \infty) = \{x \in \mathbb{R} | x \geq a\} $$
$$ (-\infty, b) = \{x \in \mathbb{R} | x < b\} $$
$$ (-\infty, b] = \{x \in \mathbb{R} | x \leq b\} $$
---
Page 406
**Definition**
**Unions and Intersections of an Indexed Collection of Sets**
Given sets $A_0, A_1, A_2, \dots$ that are subsets of a universal set $U$, and
given a nonnegative integer $n$,
$$ \bigcup_{i = 0}^{n}A_i = \{x \in U | x \in A_i \text{ for at least one } i = 0, 1, 2, \dots, n\} $$
$$ \bigcup_{i = 0}^{\infty}A_i = \{x \in U | x \in A_i \text{ for at least one nonnegative integer } i\} $$
$$ \bigcap_{i = 0}^{n}A_i = \{x \in U | x \in A_i \text{ for every } i = 0, 1, 2, \dots, n\} $$
$$ \bigcap_{i = 0}^{\infty}A_i = \{x \in U | x \in A_i \text{ for every nonnegative integer } i\} $$
---
Page 408
**Definition**
Two sets are called **disjoint** if, and only if, they have no elements in
common.
Symbolically:
$$ A \text{ and } B \text{ are disjoint } \Leftrightarrow A \cap B = \emptyset $$
---
Page 408
**Definition**
Sets $A_1, A_2, A_3, \dots$ are **mutually disjoint** (or **pairwise disjoint**
or **nonoverlapping**) if, and only if, no two sets $A_i$ and $A_j$ with
distinct subscripts have any elements in common. More precisely, for all
integers $i$ and $j = 1, 2, 3, \dots$
$$ A_i \cap A_j = \emptyset \text{ whenever } i \neq j $$
---
Page 408
**Definition**
A finite or infinite collection of nonempty sets $\{A_1, A_2, A_3, \dots\}$ is a
**partition** of a set $A$ if, and only if,
1. $A$ is the union of all the $A_i$.
2. the sets $A_1, A_2, A_3, \dots$ are mutually disjoint.
---
Page 409
**Definition**
Given a set $A$, the **power** set of $A$, denoted $\mathscr{P}(A)$, is the set
of all subsets of $A$.
---
Page 410
**Algorithm 6.1.1 Testing whether $A \subseteq B$**
_[The input sets $A$ and $B$ are represented as one-dimensional arrays
$a[1], a[2], \dots, a[m]$ and $b[1], b[2], \dots, b[n]$, respectively. Starting
with $a[1]$ and for each successive $a[i]$ in $A$, a check is made to see
whether $a[i]$ is in $B$. To do this, $a[i]$ is compared to successive elements
of $B$. If $a[i]$ is not equal to any element of $B$, then the output string,
called answer, is given the value "$A \nsubseteq B$." If $a[i]$ equals some
element of $B$, the next successive element in $A$ is checked to see whether it
is in $B$. If every successive element of $A$ is found to be in $B$, then the
answer never changes from its initial value "$A \subseteq B$."]_
**Input:** _$m$ [a positive integer], $a[1], a[2], \dots, a[m]$ [a
one-dimensional array representing the set $A$], $n$ [a positive integer],
$b[1], b[2], \dots, b[n]$ [a one-dimensional array representing the set $B$]_
**Algorithm Body:**
$i := 1, \text{answer} := A \subseteq B\\ \text{\textbf{while}} (i \leq m \text{ and answer } = A \subseteq B )\\ \ \ j := 1, \text{found} := \text{"no"}\\ \ \ \text{\textbf{while }} (j \neq n \text{ and } \text{found}= \text{"no"})\\ \ \ \ \ \text{\textbf{if }} a[i] = b[j] \text{\textbf{ then }} \text{found} := \text{"yes"}\\ \ \ \ \ j := j + 1\\ \ \ \text{\textbf{end while}}\\ \ \ \text{[If found has not been given the value "yes" when execution reaches this point, then } a[i] \neq B\text{ .]}\\ \ \ \text{\textbf{if }} \text{found} = \text{"no"} \text{\textbf{ then }} \text{answer} := A \nsubseteq B\\ \ \ i := i + 1\\ \text{\textbf{end while}}$
**Output:** _answer [a string]_

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Page 411$a
**Test Yourself**
1. The notation $A \subseteq B$ is read "_____" and means that _____.
2. To use an element argument for proving that a set $X$ is a subset of a set
$Y$, you suppose that _____ and show that _____.
3. To disprove that a set $X$ is a subset of a set $Y$, you show that there is
_____.
4. An element $x$ is in $A \cup B$ if, and only if, _____.
5. An element $x$ is in $A \cap B$ if, and only if, _____.
6. An element $x$ is in $B - A$ if, and only if, _____.
7. An element $x$ is in $A^c$ if, and only if, _____.
8. The empty set is a set with _____.
9. The power set of a set $A$ is _____.
10. Sets $A$ and $B$ are disjoint if, and only if, _____.
11. A collection of nonempty sets $A_1, A_2, A_3, \dots$ is a partition of a set
$A$ if, and only if, _____.