Question

For any two sets $A$ and $B$ if $A\cap X=B\cap X=\varphi$ and $A\cup X=B\cup X$ for some set $X$, then

• A. $A–B=A\cap B$
• B. $A=B$
• C. $B–A=A\cap B$
• D. None of these

Answer 1

The correct option is B

$A=B$

Explanation for the correct option:

Step 1. Let $A\cap (A\cup X)$ be $A$:

$\Rightarrow$ $A=A\cap (A\cup X)$

$=A\cap (B\cup X)$ $\left[\because A\cup X=B\cup X\right]$

$=(A\cap B)\cup (A\cap X)$ [By Distributive law]

$=(A\cap B)\cup \varphi$ $\left[\because A\cap X=\varphi \right]$

$=A\cap B$ ……(1)

Step 2. Let $B\cap (B\cup X)$ be $B$

$\Rightarrow$$B=B\cap (B\cup X)$

$=B\cap (A\cup X)$ $\left[\because A\cup X=B\cup X\right]$

$=(B\cap A)\cup (B\cap X)$ [By Distributive law]

$=(B\cap A)\cup \varphi$ $\left[\because B\cap X=\varphi \right]$

$=B\cap A$

$=A\cap B$ ……(2)

From equation (1) and (2), we get

$\mathit{A}\mathbf{}\mathbf{=}\mathbf{}\mathit{B}$

Hence, option ‘B’ is correct.