Question

Let $S=1,2,3,4$, then 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)

$41$

If $A$ and $B$ are disjoint sets, then $A\cap B=\varphi$.

We have to choose $A$ and $B$ such that they are disjoint subsets of $S=\{1,2,3,4\}$.

They have to be unordered pair too.

Each element in $S$ can be an element of $A$ or of $B$ or of neither subsets.

For each element, there are three possibilities.

Hence for four elements there are $3^4$ possibilities.

Now this contains ordered pairs also except for the case where both $A$ and $B$ are null sets (This appears only once).

Hence total number of ordered pairs of subsets $=3^4+1=82$

$\therefore$ total number of unordered pairs$=\frac{82}{2}=41$