Question

If $A$ and $B$ are two given sets, then
$A\cap (A\cap B{)}^c$ is equal to

• A. $A$
• B. $B$
• C. $\varnothing$
• D. $A\cap B^c$

Answer 1

The correct option is D

$A\cap B^c$

We can solve the given expression by using laws of operations on sets as follows:

$A\cap (A\cap B{)}^c$

$=A\cap (A^c\cup B^c)$ $\left[\text{De Morgan's law}\right]$

$=(A\cap A^c)\cup (A\cap B^c)$

$=\varphi \cup (A\cap B^c)$

$=A\cap B^c$