Question

If X and Y are two sets and $X^{\prime }$ denotes the complement of X, then $X\cap (X\cup Y{)}^{\prime }$ = ___.

• A. X
• B. Y
• C. $\varphi$
• D. $X\cap Y$

Answer 1

The correct option is C $\varphi$

$X\cap (X\cup Y{)}^{\prime }=X\cap (X^{\prime }\cap Y^{\prime }\left)\right[\because (A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }]=\left({X\cap X}^{\prime }\right)\cap Y^{\prime }[\because A\cap (B\cap C)=(A\cap B)\cap C]=\varphi \cap Y^{\prime }[\because A\cap A^{\prime }=\varphi ]=\varphi [\because \varphi \cap A=\varphi ]$