Need Help Simplifying Set Expression Using Set Identities

Question

one of the questions in our textbook requires us to simplify a set expression using set laws such as distributive laws, associative laws and so on. $$ ((A\cap (B\cup C))\cap (A-B))\cap (B\cup C') $$ Here's what I have so far. $$ ((A\cap (B\cup C))\cap (A\cap B'))\cap (B\cup C') $$ $$ (((A\cap B)\cup (A\cap C))\cap (A\cap B'))\cap (B\cup C') $$ $$ (((B\cap C)\cup A)\cap (A\cap B'))\cap (B\cup C') $$ $$ ((B\cap C)\cup A)\cap ((A\cap B')\cap (B\cup C') $$

Answer 1

Note that $X\cap(Y\cup Z)=(X\cap Y)\cup (X\cap Z)$, and if we for example have $X=B$, $Y=B'$, this simpliefies a lot