If A and B are two given sets, then \(A \cap (A \cap B)^c\) is equal to
(a) \(A\)
(b) \(B\)
(c) \(\Phi\)
(d) \(A \cap B^c\)
Solution
\[ A\cap(A\cap B)^c \]
Using De Morgan’s law,
\[ =A\cap(A^c\cup B^c) \]
\[ =(A\cap A^c)\cup(A\cap B^c) \]
\[ =\Phi\cup(A\cap B^c) \]
\[ =A\cap B^c \]
Answer
\[ \boxed{A\cap B^c} \]
Correct option: (d)