diff --git a/chapter_6/exercises.md b/chapter_6/exercises.md new file mode 100644 index 0000000..7f03b95 --- /dev/null +++ b/chapter_6/exercises.md @@ -0,0 +1,454 @@ +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 $a[1], a[2], \dots, a[n]$. diff --git a/chapter_6/notes.md b/chapter_6/notes.md new file mode 100644 index 0000000..51ebba6 --- /dev/null +++ b/chapter_6/notes.md @@ -0,0 +1,172 @@ +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]_ diff --git a/chapter_6/test_yourself.md b/chapter_6/test_yourself.md new file mode 100644 index 0000000..d8461f1 --- /dev/null +++ b/chapter_6/test_yourself.md @@ -0,0 +1,28 @@ +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, _____.