$A$ and $B$ are two subsets of set $S$ = $\{1,2,3,4\}$ such that $A\ \cup \ B$ = $S$ , then number of ordered pair of $(A, B)$ is 

Let $A=\{a, b\}, B=\{a, b, c\} .$ Is $A \subset B \,?$ What is $A \cup B \,?$

If $A, B$ and $C$ are any three sets, then $A - (B \cap C)$ is equal to

Let $A$ and $B$ be sets. If $A \cap X=B \cap X=\phi$ and $A \cup X=B \cup X$ for some set $X ,$ show that $A = B$

( Hints $A = A \cap (A \cup X),B = B \cap (B \cup X)$ and use Distributive law )

For any sets $\mathrm{A}$ and $\mathrm{B}$, show that

$P(A \cap B)=P(A) \cap P(B).$

Let $P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}$ and $Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}$ be two sets. Then