Question

Prove that $A-(B\cup C)=(A-B)\cap (A-C)$ for any three sets $A,B,C$.

Answer 1

To prove:

$A-(B\cup C)=(A-B)\cap (A-C)$

Solution:

L.H.S$=A-(B\cup C)$

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

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

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

$=(A-B)\cap (A-C)$

$=$ R.H.S