Question

If $X$ and $Y$ are two sets, then $X\cap (Y\cup X{)}^{\prime }$ equals

• A. $X$
• B. $Y$
• C. $\varphi$
• D. None of these

Answer 1

The correct option is C $\varphi$

$X\cap (Y\cup X{)}^{\prime }=X\cap (X^{\prime }\cap Y^{\prime })$
$(\because (X\cup Y{)}^{\prime }=X^{\prime }\cap Y^{\prime })$
$=(X\cap X^{\prime })\cap Y^{\prime }$ (by associative law)
$=\varphi \cap Y^{\prime }$ ($\because X\cap X^{\prime }=\varphi$)
$=\varphi$