Question

Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to

• A. 25
• B. 34
• C. 42
• D. 41

Answer 1

Answer: (D)

Let $A$ and $B$ be two subsets of $S$. If $x\in S$, then $x$ will not belong to $A\cap B$ if $x$ belongs to at most one of $A$, $B$. This can happen in 3 ways.
Thus, there are $3^4=81$ subsets of $S$ for which $A\cap B=\varphi$.
Out of these there is just one way for which $A=B=\varphi$.
As, we, are interested in unordered pairs of disjoint sets, the number of such subsets is
$\frac{1}{2}\left(3^4-1\right)+1=41$