# What is Power Set – Definition, Formula and Examples

Here you will learn what is power set and its definition with examples.

Let’s begin –

## What is Power Set ?

Definition : Let A be a set. Then the collection or family of all subsets of A is called the power set of A and is denoted by P(A).

That is,  P(A) = { S : S $$\subset$$ A}

Since the empty set and the set A itself are subsets of A and are therefore elements of P(A). Thus, the power set of a given set is always non-empty.

#### Power Set of Empty Set

If A is the void set $$\phi$$, then P(A) has just one element $$\phi$$ i.e $$P(\phi)$$ = {$$\phi$$}

Example 1 : Let A = {1, 2. 3}. Then, the subsets of A are :

$$\phi$$, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} and {1, 2, 3}

Hence, P(A) = { $$\phi$$, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} and {1, 2, 3} }

Example 2 : Show that n{P[P.(P($$\phi$$))]} = 4.

Solution : We have,  $$P(\phi)$$ = {$$\phi$$}

$$\therefore$$  $$P(P(\phi))$$ = {$$\phi$$, {$$\phi$$}}

$$\implies$$  $$P[P.(P(\phi))]$$ = {$$\phi$$, {$$\phi$$}, {{$$\phi$$}}, {$$\phi$$. {$$\phi$$}}}

Hence, $$P[P.(P(\phi))]$$  consists of 4 elements i.e  n{P[P.(P($$\phi$$))]} = 4

We know that a set having n elements has $$2^n$$ subsets. Therefore, if A is a finite set having n elements, then P(A) has $$2^n$$ elements.

Example 3 : If A = {a, {b}}, find P(A).

Solution : Let B = {b}. Then, A = {a, B}.

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

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