Let A be a set containing n elements. P and Q are two subsets of A. The number of ways of choosing P and Q so that

	Column I		Column II
(A)	$P\cup Q=A$	(P)	$2^n$
(B)	$P\cup Q\ne \varphi$	(Q)	$3^n$
(C)	$P\cap Q$ is a singleton set	(R)	$n\cdot 3^{n-1}$
(D)	$P\cap Q=\varphi$	(S)	$4^n-1$

Let  $A=\{a_1,a_2,\mathrm{.....}{a}_n\}$. For each  $a_i\in A$, we have following four choices
(1)  $a_i\in P,a_i\in Q$
(2)  $a_i\in P,a_i\notin Q$
(3)  $a_i\notin P,a_i\in Q$ 
(4)  $a_i\notin P,a_i\notin Q$
For  $P\cup Q=AorP\cap Q=\varphi$ the number of choice is $3^n$. The singleton can be chosen in  $n_{C_1}$ ways, rest of the elements in $3^{n-1}$ ways. $P\cup Q=\varphi$ in just one way.