Question

Show that $A\cup B=A\cap B$ implies $A=B$.

Answer 1

Step 1:
Let $x\in A\Rightarrow x\in A\cup B$
$\therefore x\in A\cap B(\text{Given}A\cup B=A\cap B)$
So, $X\in B$
$\therefore$ If $x\in A$, then $x\in B$
$\therefore A\subset B\cdots \left(i\right)$

Step 2:
Let $y\in B\Rightarrow y\in A\cup B$
$\Rightarrow y\in A\cap B(\text{Given}A\cup B=A\cap B)$
So, $y\in A$
$\therefore$ if $y\in B$, then $y\in A$
So, $B\subset A\cdots \left(ii\right)$
$\because A\subset B$ and $B\subset A$
$\Rightarrow A=B$

Final answer:
It is proved that $A=B$.