Question

Comment on the following statements
$A-B=A\cap B^{\prime }=B^{\prime }-A^{\prime }$

Answer 1

$x\in A-B\iff x\in A$

and $x\notin B\iff x\in A$ and $x\in B^{\prime }$

$\iff x\in A\cap B^{\prime }$

Hence $A-B=A\cap B^{\prime }.$ Again ....(1)

$x\in A\cap B^{\prime }\iff x\in A$ and $x\in B^{\prime }$

$\iff x\notin A^{\prime }$ and $x\in B^{\prime }\iff x\in B^{\prime }$ and $x\notin A^{\prime }$

$\iff x\in (B^{\prime }-A^{\prime })$

Hence $A\cap B^{\prime }=B^{\prime }-A^{\prime }$ ....(2)