Question

Let $A,B$ be two subsets of a Universal Set $U$,
show that $A\cap B=A-B^{\prime }=B-A^{\prime }$.

Answer 1

SInce, we know that
$A-B=A\cap B^{\prime }$
$\Rightarrow A-B^{\prime }=A\cap (B^{\prime }{)}^{\prime }=A\cap B$ $\because (A^{\prime }{)}^{\prime }=A$
Again, $\Rightarrow B-A^{\prime }=B\cap (A^{\prime }{)}^{\prime }=B\cap A=A\cap B$
$\therefore A\cap B=A-B^{\prime }=B-A^{\prime }$