GATE CS 1995 Q1.20 — Sets, Relations and Functions

To find the number of elements in the power set $P(S)$ of the set $S$, we first need to determine the cardinality (the number of elements) of $S$.

The given set is:

Let us list the distinct elements of $S$:

1. The first element is the set containing the empty set: $\{\emptyset\}$
2. The second element is the number: $1$
3. The third element is the set: $\{2, 3\}$

Since all three elements are distinct, the cardinality of $S$ is:

The power set $P(S)$ is the set of all subsets of $S$. The number of elements in the power set of any finite set $S$ is given by $2^{|S|}$.

Therefore, the number of elements in the power set $P(S)$ is $8$.