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

The correct option is D 41

S = {1, 2, 3, 4}
Let P and Q be disjoint subsets of S
Now for any element a $ϵ$ S, following cases are possible
$aϵP\text{or}aϵQ,a\notin P\text{or}aϵQ,a\notin P\text{or}a\notin Q$
$\Rightarrow$ For every element there are three options
$\therefore$ Total options $=3^4=81$
Here $P\ne Q$ except when P = Q = $\varphi$
$\therefore$ 80 ordered pairs (P, Q) are there for which $P\ne Q$. Hence total number of unordered pairs of disjoint subsets $=\frac{80}{2}+1=41$