Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

Are the two statements at the end equivalent?

share|cite|improve this question
Yes. Why not try to prove it yourself? – GEdgar Oct 1 '12 at 0:17
up vote 0 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.