TITLE: Basic set operations: Simplify (A−Bc)∪(B∩(A∩B)c).
QUESTION [1 upvotes]: Could someone please help with answering the questions below?
Question 1: Simplify $(A−B^c)∪(B∩(A∩B)^c).$
Question 2: Simplify $(A∪B)∪(C∩A)∪(A∩B)^c.$

REPLY [0 votes]: Answer for $2$nd question. (I suggest you to draw this sets, it would be much easier.) 
Since $C\cap A\subseteq A \subset A\cup B$ we have $$(A∪B)∪(C∩A)= A\cup B$$
Since $A\subset A\cup B$ and $A'\subseteq A'\cup B' = (A\cap B)'$
we have $$U = A\cup A' \subseteq (A\cup B)\cup (A\cap B)'$$ so the result is whole universal set $U$.

$1$st question.
Since $A−B^c = A\cap B$ and $B∩(A∩B)^c = B- A$ we get 
$$(A−B^c)∪(B∩(A∩B)^c) = (A\cap B)\cup (B-A) = B$$
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