Power Set

A power set includes all the subsets of a given set including the empty set. The power set is denoted by the notation P(S) and the number of elements of the power set is given by 2ⁿ. A power set can be imagined as a place holder of all the subsets of a given set, or, in other words, the subsets of a set are the members or elements of a power set.

A set, in simple words, is a collection of distinct objects. If there are two sets A and B, then set A will be the subset of set B, if all the elements of set A are present in the set B. Let us learn more about the properties of power set, the cardinality of a power set, and the power set of an empty set, with the help of examples, FAQs.

A power set is defined as the set or group of all subsets for any given set, including the empty set, which is denoted by {}, or, ϕ. A set that has 'n' elements has 2ⁿ subsets in all. For example, let Set A = {1,2,3}, therefore, the total number of elements in the set is 3. Therefore, there are 2³ elements in the power set. Let us find the power set of set A.
Set A = {1,2,3}
Subsets of set A = {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}
Power set P(A) = { {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} }

The cardinality of a set is the total number of elements in the set. A power set contains the list of all the subsets of a set. The total number of subsets for a set of 'n' elements is given by 2ⁿ. Since the subsets of a set are the elements of a power set, the cardinality of a power set is given by |P(A)| = 2ⁿ. Here, n = the total number of elements in the given set.

Example: Set A = {1,2}; n = 2
|P(A)| = 2ⁿ = 2² = 4.

Subsets of A = {}, {1},{2},{1,2}
Therefore, |P(A)| = 4.

A power set is a set that has a list of all the subsets of a given set. The power set which is denoted by P(A) with 'n' elements has the following properties:

• The total number of elements of a set is 2ⁿ.
• An empty set is a definite element of a power set.
• The power set of an empty set has only one element.
• The power set of a set with a finite number of elements is finite. For example, if set X = {b,c,d}, the power sets are countable.
• The power set of an infinite set has infinite number of subsets. For example, if Set X has all the multiples of 5 starting from 5, then we can say that Set X has an infinite number of elements. Though there is an infinite number of elements, a power set still exists for set X, in this case, it has infinite number of subsets. 
• These the power set exists for both finite and infinite sets.

Let us see how a set containing 'n' elements has a power set that has 2ⁿ elements. In other words, the cardinality of a finite set A with 'n' elements is |P(A)| = 2ⁿ.

The proof of the power set follows the pattern of mathematical induction. To start with, let us consider the case of a set with no elements or an empty set.

Case 1: A set with no elements. Let A = {}.
Here, the power set of A, which is denoted by P(A) = {} and the cardinality of the power set of A = |P(A)| = 1, since there is only one element, which is the empty set. Also, by the formula of the cardinality of a power set, there will be 2ⁿ power sets, which are equal to 2⁰ or 1.

Case 2:
This is an inductive step. It is to be proved that P(n) → P(n+1). This means, if a set that has 'n' elements has 2ⁿ subsets, then a set that has 'n+1' elements will have 2ⁿ⁺¹ subsets.
To prove this, let us assume two sets 'X' and 'Y' with the following elements.
X = {\(a_{1}\), \(a_{2}\), \(a_{3}\),\(a_{4}\), \(a_{n}\)} and
Y = {\(a_{1}\), \(a_{2}\), \(a_{3}\),\(a_{4}\), \(a_{n}\), \(a_{n+1}\)}

The cardinality of the two sets 'X' and 'Y' are,
|X| = n , which means there are 2ⁿ subsets for the set 'X'.
|Y| = n+1
We can write that Y = X U {\(a_{n+1}\)}, this means, every subset of set 'X' is also a subset of set 'Y'.
A subset of set Y may or may not contain the element \(a_{n+1}\).
If an element of set 'Y' does not contain the element \(a_{n+1}\), then it is clear that it is an element of set 'X'.
Also, if the subset of 'Y' has the element \(a_{n+1}\), this means that the element \(a_{n+1}\) is included in any of the 2ⁿ subsets of the set 'X'. So we can conclude that, set 'Y' has 2ⁿ subsets with the element \(a_{n+1}\). Therefore, set Y has 2ⁿ subsets with element \(a_{n+1}\) and 2ⁿ subsets without the element \(a_{n+1}\).
An example of this proof is as follows.
Example:
Let X = {1,2}
Let Y = {1,2,3}
Here, the |X| = 2, so there will be 2² subsets for set X.
and |Y| = 3. We will prove that set Y has 2³ subsets.
Subsets of X are = {ϕ}, {1}, {2}, {1,2}
Subsets of Y are = {ϕ}, {1}, {2}, {3}, {1,2} ,{2,3}, {1,3}, {1,2,3}
Here, '3' is the extra element in set Y that is not in set X. Also, set Y includes 4 subsets that do not include element 3 and 4 other subsets that have element 3. So, in all, for set Y there are 4 subsets without the element '3' and 4 subsets with the element '3'.

Power set of an empty set also has an element. We know that if the number of elements in a set is 'n', then there will be 2ⁿ elements in the power set. The empty set is a set with no elements. It is denoted by { } or the symbol Ø. This implies, { } is a subset of every set. An empty set does not contain any element. Therefore, the power set of the empty set is an empty set only.

We just read that an empty set does not contain any elements, this means the power set of empty set will contain 2⁰ elements. Therefore, the power set of the empty set is an empty set with one element, i.e., 2⁰ = 1. So, P(E) = {}.

☛Articles on Power Set

Given below is the list of topics that are closely connected to the Power Set. These topics will also give you a glimpse of how such concepts are covered in Cuemath

• Set Builder Notation
• Roster Notation
• Operations on Sets
• Intersection of Sets
• Subset
• Universal Set

What is a Power Set?

A set that contains all the subsets of a given set along with the empty set is called a power set. For example, if set A = {a,b}, then the power set of A is { {}, {a}, {b}, {a,b}}.

What is the Notation of a Power Set?

A power set is denoted by the letter P(Set Name). For example, if set B = {1,2,3}, then power set of B is denoted as P(B).

What is the Cardinality of a Power Set?

Cardinality denotes the total number of elements in the power set. It is denoted by |P(X)|. The cardinality of a power set for a set of 'n' elements is given by '2ⁿ'. For example, if set X = {a,b,c}, then the cardinality of the power set is |P(X)| = 2³ or 8. This means there will be 8 subsets present in the power set: { {}, {a}, {b}, {c}, {a,b}, {a, c}, {b, c}, {a, b, c} }.

What is the Power Set of an Empty Set?

A set that has no elements is said to be an empty set. A power set always has the empty set as an element. Therefore, the power set of an empty set is an empty set only. It just has one element. P(ϕ) = {ϕ}.

How Do You Find the Power Set of a Set?

To find the power set of a set, write down all the subsets of the given set along with the empty set. For example, if set S = {p,q,r}, then the power set of S, P(S) = { {}, {p},{q},{r},{p,q},{q,r},{p,r},{p,q,r} }

What are the Properties of a Power Set?

A set is a collection of distinct elements and a power set is a combination of an empty set and all the subsets of a given set. The properties of a power set are as follows.

• A power set has an empty set as an element for sure.
• The cardinality of a power set for a set of 'n' elements is given by 2ⁿ.
• The power set of an empty set has only one element which is the empty set or the null set.
• The power set of a finite set of elements is countable. For example, a set with 2 elements has 2² or 4 elements in the power set.