Question 2 of 12

Question 2

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $U=\{1,\dots,10\}$, $A=\{1,2,3,4,5\}$, $B=\{1,2,4,8\}$, $C=\{1,2,3,5,7\}$, $D=\{2,4,6,8\}$.

Find. (a) Two direct set computations. (b) Two algebraic simplifications via the laws of set theory. (c) Whether symmetric difference distributes over intersection.

Approach. Evaluate (a) directly from the element lists; simplify (b) using absorption/complement laws, then double-check against the element lists; test (c) with the given sets and, if false, exhibit a counterexample.

1. a1) $(A\cup B)\cap C$. $A\cup B=\{1,2,3,4,5,8\}$. Intersecting with $C=\{1,2,3,5,7\}$: $\boxed{(A\cup B)\cap C=\{1,2,3,5\}}$.
2. a2) $\overline{(C\cap D)}$. $C\cap D=\{2\}$ (the only element common to both). Complementing in $U$: $\boxed{\overline{C\cap D}=\{1,3,4,5,6,7,8,9,10\}}$.
3. b1) $A\cap(B-A)$. By definition $B-A=B\cap\overline{A}$ contains no element of $A$, so intersecting it back with $A$ is empty for ANY sets: $A\cap(B-A)=A\cap B\cap\overline A=(A\cap\overline A)\cap B=\varnothing\cap B=\varnothing$. $\boxed{A\cap(B-A)=\varnothing}$ (confirmed against the lists: $B-A=\{8\}$, and $A\cap\{8\}=\varnothing$).
4. b2) $(A\cap B)\cup(A\cap B\cap\overline C\cap D)\cup(\overline A\cap B)$. The middle term $A\cap B\cap\overline C\cap D$ is a subset of $A\cap B$ (absorption: $X\cup(X\cap Y)=X$), so it drops out, leaving $(A\cap B)\cup(\overline A\cap B)$. Factoring $B$ out: $(A\cup\overline A)\cap B = U\cap B = B$. $\boxed{(A\cap B)\cup(A\cap B\cap\overline C\cap D)\cup(\overline A\cap B)=B=\{1,2,4,8\}}$.
5. c) Test $A\Delta(B\cap C)=(A\Delta B)\cap(A\Delta C)$ with the given sets. $B\cap C=\{1,2\}$, so $A\Delta(B\cap C)=(A-\{1,2\})\cup(\{1,2\}-A)=\{3,4,5\}$. On the right, $A\Delta B=\{3,5,8\}$ and $A\Delta C=\{4,7\}$, so $(A\Delta B)\cap(A\Delta C)=\varnothing$. Since $\{3,4,5\}\ne\varnothing$, the identity is $\boxed{\text{FALSE}}$ for these sets (and hence false in general — a single counterexample suffices).