Questions on Intersections and Differences

Question

I was wondering whether $(A-B)\cap C=(A\cap C)-(B\cap C)$. In a Venn diagram, it seems to be true but I didn't know if there was a special case for which this is not equivalent. Thank you! (Here I am using - instead of \ for set differences as a note on my notation)

Answer 1

\begin{align} (A\cap C)-(B\cap C)&=(A\cap C)\cap(B\cap C)^c\\ &=(A\cap C)\cap (B^c\cup C^c)\\ &=(A\cap C\cap B^c)\cup(A\cap C\cap C^c)\\ &=(A\cap B^c\cap C)\cup\emptyset\\ &=(A-B)\cap C \end{align}

Answer 2

Taking an universal set (for example, $A\cup B\cup C$), we have $$(A\setminus B)\cap C=A\cap\bar B\cap\ C$$ And $$(A\cap C)\setminus(B\cap C)=(A\cap C)\cap\overline{B\cap C}=(A\cap C)\cap(\bar B\cup\bar C)=(A\cap C\cap\bar B)\cup(A\cap C\cap\bar C)$$ Since the second term of the union is empty, both set are the same.