1. **Problem Statement:** Find the specified sets and their operations for questions 9 and 10. --- ### Question 9 Given universal set $$\varepsilon = \{2,4,6,8,10,12\}$$ and subsets: $$A = \{2,6,10,12\}, B = \{4,8\}, C = \{2,4,8,10\}$$ We need to find: a. $$A'$$ (complement of A) = elements in $$\varepsilon$$ not in $$A$$ b. $$B'$$ = elements in $$\varepsilon$$ not in $$B$$ c. $$C'$$ = elements in $$\varepsilon$$ not in $$C$$ d. $$A' \cup B'$$ = union of complements of A and B e. $$B' \cap C'$$ = intersection of complements of B and C f. $$C' \cap A'$$ = intersection of complements of C and A g. $$A' \cup B'$$ (repeated, same as d) h. $$(A \cup B)'$$ = complement of union of A and B i. $$A \cap A'$$ = intersection of A and its complement (should be empty) j. $$A' \cap C$$ = intersection of complement of A and C **Step-by-step:** 1. Calculate complements: $$A' = \varepsilon - A = \{4,8\}$$ $$B' = \varepsilon - B = \{2,6,10,12\}$$ $$C' = \varepsilon - C = \{6,12\}$$ 2. Calculate unions and intersections: $$A' \cup B' = \{4,8\} \cup \{2,6,10,12\} = \{2,4,6,8,10,12\} = \varepsilon$$ $$B' \cap C' = \{2,6,10,12\} \cap \{6,12\} = \{6,12\}$$ $$C' \cap A' = \{6,12\} \cap \{4,8\} = \emptyset$$ $$(A \cup B) = \{2,4,6,8,10,12\} = \varepsilon$$ $$(A \cup B)' = \varepsilon - \varepsilon = \emptyset$$ $$A \cap A' = \{2,6,10,12\} \cap \{4,8\} = \emptyset$$ $$A' \cap C = \{4,8\} \cap \{2,4,8,10\} = \{4,8\}$$ --- ### Question 10 Given universal set $$\varepsilon = \{1,2,3,\ldots,10\}$$ and subsets: $$A = \{1,2,3,4,5\}, B = \{1,2,6,7\}, C = \{3,5\}$$ Find: a. $$n(A \cap B)$$ = number of elements in intersection of A and B b. $$n(B \cap C)$$ c. $$n(A \cup B)$$ d. $$n((A \cap B)')$$ = number of elements in complement of intersection e. $$n((B \cap C)')$$ f. $$n((A \cup B)')$$ **Step-by-step:** 1. Find intersections: $$A \cap B = \{1,2\}$$ so $$n(A \cap B) = 2$$ $$B \cap C = \emptyset$$ so $$n(B \cap C) = 0$$ 2. Find union: $$A \cup B = \{1,2,3,4,5,6,7\}$$ so $$n(A \cup B) = 7$$ 3. Complements relative to $$\varepsilon$$ (which has 10 elements): $$(A \cap B)' = \varepsilon - (A \cap B) = 10 - 2 = 8$$ $$(B \cap C)' = 10 - 0 = 10$$ $$(A \cup B)' = 10 - 7 = 3$$ --- **Final answers:** **Q9:** $$A' = \{4,8\}, B' = \{2,6,10,12\}, C' = \{6,12\}$$ $$A' \cup B' = \varepsilon, B' \cap C' = \{6,12\}, C' \cap A' = \emptyset$$ $$(A \cup B)' = \emptyset, A \cap A' = \emptyset, A' \cap C = \{4,8\}$$ **Q10:** $$n(A \cap B) = 2, n(B \cap C) = 0, n(A \cup B) = 7$$ $$n((A \cap B)') = 8, n((B \cap C)') = 10, n((A \cup B)') = 3$$