Question

Let $S$ be a set containing n elements and we select two subsets $A$ and $D$ of S at random, then the probability that $A \cup B = S$ and $A \cap B = \phi$ is:

• A. $2^n$
• B. $n^2$
• C. $\frac{1}{n}$
• D. $\frac{1}{2^n}$

Answer 1

Answer: (D)

Given that $S$ contains $n$ elements and two sets $A$ and $B$ are selected.
Two sets $A$ and $B$ can be selected in $2^{ n }$ ways.
The number of ways of selecting two sets such that their union is $S$ and intersection is nullset is $1$ .
Therefore the probability is $\frac{1}{2^{n}}$.