# How to know which of these sets are equivalent

Let $A, B, C$ be sets that are all contains in a universal group $U$. 3 out of 4 of 4 of these groups are equivalent. Which one isn't necessarily equivalent?

A. $((A \cap B) \cup (B \cap C)) \cap (\bar{A} \cup \bar{C})$

B. $(A \cap B) \triangle (C \cap B)$

C. $((A \cap B) \setminus C) \cup ((B \cap C) \setminus A)$

D. $(A \cap B) \setminus ( A \cap C)$

Where $\bar{A}$ is A's complement and $\triangle$ is symmetric difference

I tried solving the question with venn diagrams but I couldn't really get this solved.

Can anyone please recommend an approach of solving this? I believe to have a good understanding of set operations but still having trouble solving this.

Thanks!

Then $A\cap B=\{a\}$ and $B\cap C= \{a, d\}$ so $(A\cap B)\cup (B\cap C)= \{a, d\}$. $\overline{A}= \{d, e\}$ and $\overline{C}= \{c, e\}$ so that $\overline{A}\cup \overline{C}= \{c, d, e\}$. The intersection with the previous set is $\{d\}$. Do the same with each of the others.