Question

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

Answer 1

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

Answer 2

if A=B so the equation becomes :

A union A = A intersection A

now A union A = A
A intersection A = A
..

Answer 3

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.

Answer 4

The easy way goes like this.

If A = B, then

\[A \cup B = A \cup A = A \]

Now

\[A = A \cap A = A \cap B\]