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Let $A$ and $B$ each be a set and assume that $A \cup B \subseteq A \cap B$. Prove that $A = B$.

Proof: Let $x$ be an element of $A$. Therefore $x$ is an element of $A \cup B$. Since $A \cup B$ is a subset of $A \cap B$, $x$ is an element of $A \cap B$. Therefore, $x$ is an element of $B$ also. Therefore, $A$ is a subset of $B$. Using the same argument with $A$ and $B$ switched around we find that $B$ is a subset of $A$. Thus $A = B$.

This proof seems kind of simple. Could someone check it for me and let me know if it is correct or not?

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It’s just fine. Sometimes things are simple. :-) – Brian M. Scott Dec 10 '12 at 6:49
What's wrong with simple proofs? Simple proofs are nice :) This works, in fact I can't really imagine doing this any other way. – Gyu Eun Lee Dec 10 '12 at 6:49
Awesome. I figured since it was short that I may have missed something. Thanks. – blutuu Dec 10 '12 at 6:50
up vote 0 down vote accepted

As other stated, the solution is correct. This answer (which I'm making CW) is to just so the question has an official answer.

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Ok cool. I have another question that doesn't have an answer, but has reached a conclusion. – blutuu Dec 10 '12 at 7:06
@blutuu: Just as an FYI: It's permissible for you to add your own answer to a question, too. You could do that for your other question. – JavaMan Dec 10 '12 at 7:09
I started to do that, but I wasn't sure if I should answer it or not. I'll keep that in mind. – blutuu Dec 10 '12 at 7:14

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