1. **Determine True or False for each statement:** - a. $a \in A$ (Cannot determine without set $A$ definition) - b. $x \in A$ (Cannot determine without set $A$ definition) - c. $b \in A$ (Cannot determine without set $A$ definition) - d. $e \in c$ (Likely false, $e$ is an element, $c$ is a set) - e. $f$ (Incomplete statement) - f. Set $A$ is a finite set (Cannot determine without $A$ definition) 2. **Pick the correct answer:** - a. $N = \{1, 2, 3, 4, \ldots\}$ is an infinite set. - b. $A = \emptyset$ is a null set. - c. $5 \in \{3, 8, 9, 14\}$ is False. - d. $x = \frac{4}{7} + 3 \in \{x : x \text{ is an integer}\}$ is False because $\frac{4}{7} + 3 = \frac{25}{7}$ is not an integer. 3. **Write sets in roster form:** - a. $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ - b. $B = \{w, x, y, z\}$ - c. $C = \{11, 13, 15, 17, 19, 21, 23\}$ - d. $D = \{2, 4, 6, 8, 10\}$ - e. $E = \{12, 14, 15, 16, 18\}$ (five non-prime numbers greater than 10) - f. $F = \{42, 43, 44, 45, 46, 47, 48, 49\}$ - g. $G = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\}$ - h. $H = \{April, August\}$ - i. $I = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$ 7. Given $M = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$ and $N = \{3, 6, 9, 11, 13\}$: - a. $M \cap N = \{3, 9, 11, 13\}$ - b. $M \cup N = \{1, 3, 5, 6, 7, 9, 11, 13, 15, 17, 19\}$ 8. Given $P = \{1, 2, \ldots, 49\}$, $Q = \{5, 10, 15, 20, 25, 30, 35, 40, 45\}$, $R = \{2, 4, 6, 8, 10\}$: - a. $P \cap Q = Q$ - b. $P \cap R = R$ - c. $Q \cap R = \{10\}$ - d. $P \cap Q \cap R = \{10\}$ - e. $(P \cup R) \cap Q = Q$ - f. $(P \cap Q) \cup R = Q \cup R = \{2, 4, 5, 6, 8, 10, 15, 20, 25, 30, 35, 40, 45\}$ 9. Given universal set $\varepsilon = \{2, 4, 6, 8, 10, 12\}$, $A = \{2, 6, 10, 12\}$, $B = \{4, 8\}$, $C = \{2, 4, 8, 10\}$: - a. $A' = \varepsilon \setminus A = \{4, 8\}$ - b. $B' = \varepsilon \setminus B = \{2, 6, 10, 12\}$ - c. $C' = \varepsilon \setminus C = \{6, 12\}$ - d. $A' \cup B' = \{4, 8\} \cup \{2, 6, 10, 12\} = \varepsilon$ - e. $B' \cap C' = \{2, 6, 10, 12\} \cap \{6, 12\} = \{6, 12\}$ - f. $C' \cap A' = \{6, 12\} \cap \{4, 8\} = \emptyset$ - g. $A' \cup B' = \varepsilon$ - h. $(A \cup B)' = \varepsilon \setminus (A \cup B) = \varepsilon \setminus \varepsilon = \emptyset$ - i. $A \cap A' = \emptyset$ - j. $A' \cap C = \{4, 8\} \cap \{2, 4, 8, 10\} = \{4, 8\}$ 10. Given $\varepsilon = \{1, 2, \ldots, 10\}$, $A = \{1, 2, 3, 4, 5\}$, $B = \{1, 2, 6, 7\}$, $C = \{3, 5\}$: - a. $n(A \cap B) = n(\{1, 2\}) = 2$ - b. $n(B \cap C) = n(\emptyset) = 0$ - c. $n(A \cup B) = n(\{1, 2, 3, 4, 5, 6, 7\}) = 7$ - d. $n((A \cap B)') = n(\varepsilon \setminus \{1, 2\}) = 8$ - e. $n((B \cap C)') = n(\varepsilon \setminus \emptyset) = 10$ - f. $n((A \cup B)') = n(\varepsilon \setminus \{1, 2, 3, 4, 5, 6, 7\}) = 3$