Question

For any three sets , A B and C, $B\setminus (A\cup C)$ is:

• A. $(A\setminus B)\cap (A\setminus C)$
• B. $(B\setminus A)\cap (B\setminus C)$
• C. $(B\setminus A)\cap (A\setminus C)$
• D. $(A\setminus B)\cap (B\setminus C)$

Answer 1

The correct option is B $(B\setminus A)\cap (B\setminus C)$

In the problem statement we are taking union of $A$ and $C$ and then taking its difference from $B$

This means $B$ \ $(A\cup C)$ will contain elements that are in $B$ but not in both $B$ $\text{and}$ $C$.

$\therefore B$ \ $(A\cup C)$ is equivalent to taking difference of $A,B$ and $A,C$ and then taking there intersection i.e. $(B$ \ $A)\cap (B$ \ $C)$

Hence option ($b$) is correct.