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If $A\cap B = B\cap C$, then $(A-B)\cup C = A\cup (C-B)$.

Are the two statements at the end equivalent?

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Yes. Why not try to prove it yourself? –  GEdgar Oct 1 '12 at 0:17
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Suppose that $x\in(A-B)\cup C$.

If $x\in C$ and $x\notin B$ then $x\in C-B$ so $x\in A\cup(C-B)$.

If $x\in C$ and $x\in B$ then $x\in B\cap C=A\cap B$ so $x\in A$. Therefore $x\in A\cup C-B$.

If $x\in A-B$ then $x\in A$ so $x\in A\cup C-B$.

This shows the first inclusion. The statement is symmetric in $A$ and $C$ so switching the role of the two shows inclusion in the other direction.

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