Question

Prove that $A\cup B=A\cap B$, if $A=B$.

Answer 1

$P\left(A\right)=P\left(B\right)$

inorder to prove $A=B$, we should prove $A$ is a subset of $B$ i.e., $A\subset B$

and $B$ is a subset of $A$ i.e., $B\subset A$

Set $A$ is an element of power et of $A$ as every set is a subset

i.e., $P\in P\left(A\right)$

$A\in P\left(B\right)$

if set $A$ is in power set of $B$

Set $A$ is a subset of $B$

$A\subset B$

Similarly we can prove

$B\subset A$

Now since $A\subset B$ and $B\subset A$

$\therefore$ $A=B$

Hence proved