If $U = \{a, b, c, d, e, f\}$,$A = \{a, b, c\}$,$B = \{c, d, e, f\}$,and $C = \{c, d, e\}$,find $(A \cap B) \cup (A \cap C)$.

If $n(U) = 600$,$n(A) = 100$,$n(B) = 200$,and $n(A \cap B) = 50$,then $n(\bar{A} \cap \bar{B})$ is (where $U$ is the universal set and $A$ and $B$ are subsets of $U$).

In a class of $55$ students,the number of students studying different subjects are $23$ in Mathematics,$24$ in Physics,$19$ in Chemistry,$12$ in Mathematics and Physics,$9$ in Mathematics and Chemistry,$7$ in Physics and Chemistry and $4$ in all the three subjects. The number of students who have taken exactly one subject is

Let $n(U) = 700, n(A) = 200, n(B) = 300$ and $n(A \cap B) = 100,$ then $n(A^c \cap B^c) = $

The domain of the function $f(x) = \sqrt{\log \frac{1}{|\sin x|}}$ is

The domain of the function $f(x) = \log_{3 + x}(x^2 - 1)$ is