Question

Prove that : $A-(B\cup C)=(A-B)\cap (A-C)$ without using venn diagram.

Answer 1

If $A-(B\cap C)=\varphi$ then the result holds by properties of $\varphi$. So assume

$A-(B\cap C)\ne \varphi$ and let $x\in A-(B\cap C)$

The $x\in A$ and $x\notin B\cap C$ which implies that $x\in A$ and $(x\notin B,x\notin C)$

Case 1:

$x\notin B$

if $x\in C$, then $x\in A-C$ which is $x\in (A-B)\cup (A-C)$

Case 2

$x\in B$

If $x\notin C$, then $x\in A-C$ which means

$x\in (A-B)\cup (A-C)$

Both cases lead to the same conclusion

$x\in (A-B)\cup (A-C)$

Hence

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

Hence proved