Power set is formed of all possible subsets of a given set. It is denoted as P(A).

- Definition 2: Power set
The collection of all subsets of a set “A” is called power set, P(A).

For example, consider a set given by :

A
=
{
1,
3,
4
}
A
=
{
1,
3,
4
}

What are the possible subsets? There are three subsets consisting of individual elements: {1}, {3} and {4}. Then, elements taken two at a time form following subsets : {1,3}, {1,4} and {3,4}. Since order or sequence does not matter in set representation, there are only three subsets of two elements taken together. Now, the elements taken three at a time form the only one subset : {1,3,4}. Remember, a set is a subset of itself. Further, empty set (φ) is subset of any set. Hence, φ is also a subset of the given set “A”.

The set comprising of all possible subsets of given set “A” is :

P
A
=
{
φ
,
{
1
}
,
{
3
}
,
{
4
}
,
{
1,3
}
,
{
1,4
}
,
{
3,4
}
,
{
1,3,4
}
}
P
A
=
{
φ
,
{
1
}
,
{
3
}
,
{
4
}
,
{
1,3
}
,
{
1,4
}
,
{
3,4
}
,
{
1,3,4
}
}

We note two important points from this representation of power set :

1: The elements of a power set are themselves sets. In other words, every element of a power set is a set.

2: If the numbers of elements (cardinality) in a set is “n”, then numbers of elements in power set is
2
n
2
n
.

For a set having three elements, the total numbers of elements in the power set is :

⇒
m
=
2
n
=
2
3
=
8
⇒
m
=
2
n
=
2
3
=
8

We can see that this result is consistent with the illustration given above. We should, here, emphasize to avoid confusion that counting of elements of a set (cardinality) excludes empty set. It is, however, counted as members of power set.