Q.If $A = \{a, b, c\}$, list all the subsets of $A$ and verify that the total number of subsets is $2^3$.

$P(\phi)=\{\phi\}$, which has exactly $1$ element.

For any set $A$ with $n$ elements, the power set $P(A)$ (the set of all subsets of $A$, including $A$ itself and the empty set) has $2^n$ elements.

Here $A=\phi$, so $n=0$. Thus $P(A)$ has $2^0=1$ element. This single element is $\phi$ itself, since the empty set is always a subset of every set (including itself): $P(\phi)=\{\phi\}$.

$n[P(A)] = 2^{n(A)}$; with $A=\phi$, $n(A)=0$, so $n[P(A)]=2^0=1$.

The power set $P(A)$ of a set $A$ is the set of all subsets of $A$, and for any set with $n(A)=n$ elements, $n[P(A)]=2^n$.

Here $A=\phi$, so $n(A)=0$. The only subset of the empty set is the empty set itself, i.e. $P(A)=\{\phi\}$.

Therefore $n[P(A)] = 2^{0} = 1$.

Taking $A=\varnothing$ (the empty set) — the standard form of this question — the power set $P(A)=\{\varnothing\}$ has exactly $1$ element.

The power set $P(A)$ of a set $A$ is the set of all subsets of $A$; if $A$ has $n$ elements, $P(A)$ has $2^n$ elements.

This question (a well-known NCERT Sets exercise) asks for the number of elements of $P(A)$ when $A$ is the empty set, $A=\varnothing$ — the printed stem's "$Q$" is almost certainly an OCR misread of the empty-set symbol $\varnothing$. The empty set has $n=0$ elements.

Its only subset is itself: $P(\varnothing) = \{\varnothing\}$.

So $P(A)$ has $2^0 = 1$ element.