1. **Problem Statement:** We are given several sets and asked to express set relations using symbols, analyze subset relations, and find intersections and unions of sets. 2. **Set Relations Symbols:** - a. X is equal to Y: $X = Y$ - b. A and B are equivalent sets: $A \equiv B$ - c. B is a null set: $B = \emptyset$ - d. J is a subset of K: $J \subseteq K$ - e. B is not equal to A: $B \neq A$ - f. A and B are disjoint sets: $A \cap B = \emptyset$ - g. x is not a member of N: $x \notin N$ - h. K is not a subset of L: $K \not\subseteq L$ 3. **Set Definitions:** - $A = \{\text{letters in "sow"}\} = \{s, o, w\}$ - $B = \{\text{letters in "vowels"}\} = \{v, o, w, e, l, s\}$ - $C = \{\text{letters in "wolves"}\} = \{w, o, l, v, e, s\}$ 4. **Check subset relations:** - a. $A \subset B$? Since $A = \{s,o,w\}$ and $B = \{v,o,w,e,l,s\}$, all elements of $A$ are in $B$, so **True**. - b. $B \subseteq A$? No, $B$ has elements not in $A$, so **False**. - c. $A \subset C$? $C = \{w,o,l,v,e,s\}$ contains all elements of $A$, so **True**. - d. $C \subseteq B$? $C$ and $B$ have the same elements, so **True**. - e. $C \subset A$? No, $C$ has elements not in $A$, so **False**. - f. $B \supset A$? Equivalent to $A \subset B$, so **True**. 5. **Calculate cardinalities:** - a. $n(A \cap B) = n(\{s,o,w\} \cap \{v,o,w,e,l,s\}) = n(\{s,o,w\}) = 3$ - b. $n(B \cap C) = n(\{v,o,w,e,l,s\} \cap \{w,o,l,v,e,s\}) = n(\{v,o,w,e,l,s\}) = 6$ - c. $n(A \cup B')$ where $B'$ is complement of $B$ in universal set (not defined), so cannot determine exact number without universal set. - d. $n(A \cap B')$ similarly undefined without universal set. - e. $n(B' \cap C')$ undefined without universal set. - f. $n(A \cup B')$ same as c. 6. **Set X, Y, and operations:** - $x = \{x : x \text{ odd integer}, 1 \leq x \leq 25\} = \{1,3,5,7,9,11,13,15,17,19,21,23,25\}$ - $X = \{x : x \text{ multiple of 3}\} = \{3,6,9,12,15,18,21,24\}$ - $Y = \{x : x \text{ multiple of 5}\} = \{5,10,15,20,25\}$ - a. $X \cap Y = \{15\}$ - b. $X \cap Y' = X \setminus Y = \{3,6,9,12,18,21,24\}$ 7. **Set P, Q, and operations:** - $x = \{x : x \text{ prime}, x < 50\} = \{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47\}$ - $P = \{x : 10 < x < 42\} = \{11,13,17,19,23,29,31,37,41\}$ - $Q = \{x : 20 < x < 50\} = \{23,29,31,37,41,43,47\}$ - a. $P \cap Q = \{23,29,31,37,41\}$ - b. $P \cup Q = \{11,13,17,19,23,29,31,37,41,43,47\}$ - c. $n(P' \cap Q')$ where $P'$ and $Q'$ are complements in $x$: - $P' = x \setminus P = \{2,3,5,7,43,47\}$ - $Q' = x \setminus Q = \{2,3,5,7,11,13,17,19,41\}$ - $P' \cap Q' = \{2,3,5,7\}$ so $n(P' \cap Q') = 4$ - d. $n(P' \cup Q) = n(x \setminus P \cup Q) = n(x) = 15$ (since $P' \cup Q$ covers all primes less than 50) **Final answers:** - Set relations symbols as above. - True/False subset relations as above. - Cardinalities and set members as above.