Set Operations 54C940

1. **Problem 14:** Find $[A \cap (B \cup U)] \cap [B \cap (C \cap U)]$ given the sets: - $U = \{a,b,c,\ldots,z\}$ (all English letters) - $A = \{t,r,i,a,n,g,l,e,s\}$ - $B = \{s,q,u,a,r,e\}$ - $C = \{h,e,x,a,g,o,n,s\}$ 2. **Step 1:** Simplify $B \cup U$. Since $U$ contains all English letters, $B \cup U = U$. 3. **Step 2:** Calculate $A \cap (B \cup U) = A \cap U$. Since $U$ is the universal set of letters, $A \cap U = A = \{t,r,i,a,n,g,l,e,s\}$. 4. **Step 3:** Simplify $C \cap U$. Again, $C \cap U = C = \{h,e,x,a,g,o,n,s\}$. 5. **Step 4:** Calculate $B \cap (C \cap U) = B \cap C$. Find common elements of $B$ and $C$: $B = \{s,q,u,a,r,e\}$ $C = \{h,e,x,a,g,o,n,s\}$ Common elements: $\{a,e,s\}$. 6. **Step 5:** Now find $[A \cap (B \cup U)] \cap [B \cap (C \cap U)] = A \cap (B \cap C)$. We have $A = \{t,r,i,a,n,g,l,e,s\}$ and $B \cap C = \{a,e,s\}$. Common elements are $\{a,e,s\}$. 7. **Answer for 14:** $\{a,e,s\}$ which corresponds to option (b). --- 8. **Problem 15:** Find $n[(A \cup B) \times (B \cup C)]$. 9. **Step 1:** Calculate $A \cup B$. $A = \{t,r,i,a,n,g,l,e,s\}$ $B = \{s,q,u,a,r,e\}$ Union combines all unique elements: $A \cup B = \{t,r,i,a,n,g,l,e,s,q,u\}$ Count elements: $t, r, i, a, n, g, l, e, s, q, u$ = 11 elements. 10. **Step 2:** Calculate $B \cup C$. $B = \{s,q,u,a,r,e\}$ $C = \{h,e,x,a,g,o,n,s\}$ Union: $s, q, u, a, r, e, h, x, g, o, n$ = 11 elements. 11. **Step 3:** The Cartesian product $(A \cup B) \times (B \cup C)$ has size: $$n[(A \cup B) \times (B \cup C)] = n(A \cup B) \times n(B \cup C) = 11 \times 11 = 121$$ 12. **Answer for 15:** 121 which corresponds to option (b).