Get rid of difference $A\setminus D$ in algebra of sets

Question

I need to simplify $[[(A\cap D)\cup(A\setminus D)]\cap(A\cup C)]\cup[B\cap(B\cap C)'],$ but I don't know how to work with the $A\setminus D$. Please help.

Answer 1

I would just use $A\setminus D=A\cap D'.$ Then it looks like you've got a lot of distribution to do.

Answer 2

$A\cap D$ is the set of members of $A$ that belong to $D.$

$A\backslash D$ is the set of members of $A$ that don't belong to $D.$

So their union $(A\cap D)\cup (A\backslash D)$ is the set of all members of $A,$ which is the set $A$ itself.