Question

If $A,B$ and $C$ are sets, then prove that $(A-B)\cap (A-C)=A-(B\cup C)$.Verify the above result by Venn diagrams.

Answer 1

Consider,

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

$=(A\cap B^{\prime })\cap (A\cap C^{\prime })$ (By def of difference of sets, $A-B=A\cap B^{\prime }$ )

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

$=A\cap (B\cup C{)}^{\prime }$ (Demorgan's law , $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$)

$=A-(B\cup C)$ (By def of difference of sets, $A-B=A\cap B^{\prime }$ )

Hence, proved.

From , the Venn diagram , its clear that common portion of $A-B$ and $A-C$ is the checks portion.

Also, it represents $A-(B\cup C)$

Hence, verified.