If A = {a, b, c}, find P(A) and n{P(A)}.
Question:

If A = {a, b, c}, find P(A) and n{P(A)}.

Solution:

The collection of all subsets of a set A is called the power set of A. It is denoted by P(A).

Now, We know that $\phi$ is a subset of every set. So, $\phi$ is a subset of $\{a, b, c\}$.

Also, $\{a\},\{b\},\{c\},\{a, b\},\{b, c\},\{a, c\}$ are also subsets of $\{a, b, c\}$

We know that every set is a subset of itself. So, $\{a, b, c\}$ is a subset of $\{a, b, c\}$.

Thus, the set $\{a, b, c\}$ has, in all eight subsets, viz. $\phi,\{a\},\{b\},\{c\},\{a, b\},\{b, c\},\{a, c\},\{a, b$, c\}.

$\therefore P(A)=\{\phi,\{a\},\{b\},\{c\},\{a, b\},\{b, c\},\{a, c\},\{a, b, c\}\}$

Now, $\mathrm{n}\{\mathrm{P}(\mathrm{A})\}=2^{\mathrm{m}}$, where $\mathrm{m}=\mathrm{n}(\mathrm{A})=3$

$\Rightarrow \mathrm{n}\{\mathrm{P}(\mathrm{A})\}=2^{3}=8$