Question

If $A$ and $B$ are subsets of a set $X,$ then $(A\cap (X-B))\cup B$ is equal to

• A. $A\cup B$
• B. $A\cap B$
• C. $A$
• D. $B$

Answer 1

The correct option is A $A\cup B$

Given, $A$ and $B$ are subsets of a set $X$.
$\Rightarrow A\subset X$ and $B\subset X$
Now, $(A\cap (X-B))\cup B$
$=(A\cap (X\cap B^{\prime }))\cup B$
$=((A\cap X)\cap B^{\prime })\cup B$
$=(A\cap B^{\prime })\cup B$
$=(A\cup B)\cap (B^{\prime }\cup B)$
$=(A\cup B)\cap U$, where $U$ is the universal set.
$=A\cup B$