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}}}.

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