A is a set containing $n$  elements. $A$ subset$P$ of $A$  is chosen. The set $A$  is reconstructed by replacing the elements of $P.$  $A$  subset $Q$  of $A$ is again chosen. The number of ways of choosing $PandQ$  so that $P\cap Q=\varphi$  is

Let $A=\left\{a_1,a_2,\mathrm{.......},a_n\right\}.$  For $a_i\in A$  , we have the following choices

    $\left(i\right)a_i\in Pand{a}_i\in Q$                             $\left(ii\right)a_i\in Pand{a}_i\notin Q$

   $\left(iii\right)a_i\notin Pand{a}_i\in Q$                              $\left(iv\right)a_i\notin Pand{a}_i\notin Q$

Out of these only (ii),(iii) and (iv) imply $a_i\notin P\cap Q.$

Therefore, the number of required subsets is $3^n.$