1. **Problem Statement:** We are given sets and asked to represent sets with symbols, analyze set relations, and find intersections, unions, and complements of sets. 2. **Set Representation Symbols:** - Equal sets: $X = Y$ - Equivalent sets: $A \equiv B$ - Null set: $B = \emptyset$ - Subset: $/ \subseteq K$ - Not equal: $B \neq A$ - Disjoint sets: $A \cap B = \emptyset$ - Not a member: $x \notin N$ - Not a subset: $K \not\subseteq L$ 3. **Set Memberships:** - $H = \{\text{months beginning with A}\} = \{\text{April, August}\}$ - $/ = \{1,3,5,7,9,11,13,15,17,19\}$ (odd numbers less than 20) - $J = \{3,6,9,12,15,18\}$ (positive multiples of 3 less than 20) 4. **Set Letters:** - $A = \{s,o,w\}$ - $B = \{v,o,w,e,l,s\}$ - $C = \{w,o,l,v,e,s\}$ 5. **True/False Statements:** - a. $A \subset B$? No, because $A$ has $s,o,w$ and $B$ has $v,o,w,e,l,s$; $A$ is a subset of $B$ since all elements of $A$ are in $B$. - b. $B \subseteq A$? No, $B$ has letters not in $A$. - c. $A \subset C$? Yes, all letters of $A$ are in $C$. - d. $C \subseteq B$? Yes, $C$ and $B$ have the same letters. - e. $C \subset A$? No. - f. $B \supset A$? Yes, $B$ contains all elements of $A$. 6. **Cardinalities:** - a. $n(A \cap B) = n(\{s,o,w\} \cap \{v,o,w,e,l,s\}) = n(\{s,o,w\}) = 3$ - b. $n(A \cup B) = n(\{s,o,w,v,e,l\}) = 6$ - c. $n(B \cap C) = n(\{v,o,w,e,l,s\} \cap \{w,o,l,v,e,s\}) = 6$ - d. $n(B \cup C) = 6$ - e. $n(A \cap B') = n(\{s,o,w\} \cap \{\text{letters not in } B\}) = 0$ - h. $n(A' \cup B') = n(\text{letters not in } A \cup \text{letters not in } B) = \text{depends on universal set}$ 7. **Set X, Y, and operations:** - $x = \{1,3,5,7,9,11,13,15,17,19,21,23,25\}$ (odd integers 1 to 25) - $X = \{3,6,9,12,15,18,21,24\}$ (multiples of 3) - $Y = \{5,10,15,20,25\}$ (multiples of 5) - a. $X \cap Y = \{15\}$ - b. $X \cap Y' = X \setminus Y = \{3,6,9,12,18,21,24\}$ 8. **Prime numbers and sets P, Q:** - $x = \{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47\}$ (primes less than 50) - $P = \{11,13,17,19,23,29,31,37,41\}$ (primes between 10 and 42) - $Q = \{23,29,31,37,41,43,47\}$ (primes between 20 and 50) - 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. $n(P' \cap Q') = n(\text{primes} \setminus P \cap \text{primes} \setminus Q) = n(\{2,3,5,7,43,47\} \cap \{2,3,5,7,11,13,17,19\}) = n(\{2,3,5,7\}) = 4$ - d. $n(P' \cup Q) = n(\text{primes} \setminus P \cup Q) = n(\{2,3,5,7,43,47\} \cup \{23,29,31,37,41,43,47\}) = 10$