Let A and B be sets. Prove that $$A \cup B = A \cap B$$ if and only if $$A=B$$

Question

jamesj · Answer

So one way is easy.  If A = B then ....

anonymous · Answer

if A=B so the equation becomes :

A union A = A intersection A

now A union A = A
A intersection A = A
..

jamesj · Answer

Now the other way.

$$A \cup B = A \cap B \implies  \forall \ x \in A \cup B , x \in A \cap B  $$
hence 
$$x \in A \hbox{ and } \ x \in B$$
Now as x was arbitrary, that means every a in A is also in B
i.e.,
$$A \subset B$$
and conversely every b in B is also in A
i.e.,
$$B \subset A$$
Therefore A = B.

jamesj · Answer

The easy way goes like this. 

If A = B, then

$$A \cup B = A \cup A = A $$

Now

$$A = A \cap A = A \cap B$$