Question

Prove that $A-(A-B)=A\cap B$

Answer 1

Consider, $A-(A-B)$

$=A-(A\cap B^{\prime })$ ($\because A-B=A\cap B^{\prime }$)

$=A\cap (A\cap B^{\prime }{)}^{\prime }$

$=A\cap (A^{\prime }\cup (B^{\prime }{)}^{\prime })$ $\because (A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$

$=A\cap (A^{\prime }\cup B)$

$=(A\cap A^{\prime })\cup (A\cap B)$ (Distributive property)

$=\varphi \cup (A\cap B)$

$=A\cap B$