The power set is a set which includes all the subsets including the empty set and the original set itself. It is usually denoted by P. Power set is a type of sets, whose cardinality depends on the number of subsets formed for a given set. If set A = {x, y, z} is a set, then all its subsets {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z} and {} are the elements of power set, such as:

Power set of A, P(A) = { {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}, {} }

Where P(A) denotes the power set.

Let us understand the concept with the help of examples and properties. 

What is a Power Set?

In set theory, the power set (or power set) of a Set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set. It is denoted by P(A). Basically, this set is the combination of all subsets including null set, of a given set.

How is Power set Calculated?

If the given set has n elements, then its Power Set will contain 2ⁿ elements. It also represents the cardinality of the power set.

Example of Power Set

Let us say Set A = { a, b, c }

Number of elements: 3

Therefore, the subsets of the set are:

• { } which is the null or the empty set
• { a }
• { b }
• { c }
• { a, b }
• { b, c }
• { c, a }
• { a, b, c }

The power set P(A) = { { } , { a }, { b }, { c }, { a, b }, { b, c }, { c, a }, { a, b, c } }

Now, the Power Set has 2³ = 8 elements.

Cardinality of Power Set

Cardinality represents the total number of elements present in a set. In case of power set, the cardinality will be the list of number of subsets of a set. The number of elements of a power set is written as |P (A)|, where A is any set. If A has ‘n’ elements then the formula to find the number of subsets of a set in a power set is given by:

|P(A)| = 2ⁿ

For example, set A = {1, 2, 3}

n = number of elements of A = 3

So, the number of subsets in a power set of A will be:

Subsets of A = {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3,}

P|A| = 2³ = 8

Hence, P(A) is {{}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3,}}

Properties of Power Set

• It is much larger than the original set.
• The number of elements in the power set of A is 2ⁿ, where n is the number of elements in set A
• The power set of a countable finite set is countable.
• For a set of natural numbers, we can do one-to-one mapping of the resulted set, P(S), with the real numbers.
• P(S) of set S, if operated with the union of sets, the intersection of sets and complement of sets, denotes the example of Boolean Algebra.

Power Set of Empty Set

An empty set has zero elements. Therefore, the power set of an empty set { }, can be mentioned as;

• A set containing a null set.
• It contains zero or null elements.
• The empty set is the only subset.

Recursive Algorithm of Power Set

A recursive algorithm is used to generate the power set P(S) of any finite set S.

The operation F (e, T) is defined as:

F (e, T) = { X ∪ {e} | X ∈ T }

This returns each of the set X in T that has the element x.

If Set S = { }, then P(S) = { { } } is returned.

If not, the following algorithm is followed.

If e is an element in Set S, T = S {e} such that S { e } forms the relative complement of the element e in set S, the power set is generated by the following algorithm:

P(S) = P(T) ∪ F ( e, P(T))

To conclude, if the set S is empty, then the only element in the power set will be the null set. If not, the power set will become the union of all the subsets containing the particular element and the subsets not containing the particular element.

How is Power-Set Related to Binomial Theorem

It is closely related to the binomial theorem in terms of the notation.

Let us consider a set of three elements S = {a, b, c}

Number of subsets with zero elements (the null or the empty set) = 1

Number of subsets with one element (the singleton subsets) = 3

Number of subsets with two elements (the complements of singleton subsets) = 3

Number of subsets with three elements (the actual set) = 1

From the above relationship we can calculate |2^s| as follows:

|2^s| = \(\sum_{k=0}^{|s|}(^{|s|}_{k})\)

If |S| = n then,

|2^s| = 2ⁿ = \(\sum_{k=0}^{n}(^{n}_{k})\)

This is the relationship between a power-set and the binomial theorem.

Problems and Solutions on Power Set

Q.1: Find the power set of Z = {2, 7, 9} and a total number of elements.

Solution: Given, Z = {2, 7, 9}

Total number of elements in power set = 2ⁿ

Here, n = 3 (number of elements in set Z)

So, 2³ = 8, which shows that there are eight elements of power set of Z

Therefore, 

P(Z) = {{}, {2}, {7}, {9}, {2, 7}, {7, 9}, {2, 9}, {2, 7, 9}}

Q.2: How many elements are there for the power set of an empty set?

Solution: An empty set has zero elements.

Therefore, no. of elements of power set = 2⁰ = 1

Hence, there is only one element of the power set which is the empty set itself.

P(E) = {}

Q.3: What is the power set of set A = {1, 2, 3, 4}?

Solution: Given, set A = {1, 2, 3, 4}

By the formula of power set, we know that, the number of sets we can form here is given by:

|P (A)| = 2ⁿ

where n is the number of elements of set A

Hence,

|P(A)| = 2⁴ = 2 x 2 x 2 x 2 = 16

There will be 16 subsets of set A.

Subsets of A = {}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1,2,3,4}.

Thus, the power set of set A is given by:

P(A) = { {}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1,2,3,4} }.

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