How to Find the Power Set of a Given Set
All of the subsets of a particular set, including the empty set, is known as a power set.
P(Set name) is the notation for the power set.
\[2^{n}\] is the number of elements in the power set.
A set is a collection of different objects. If two sets A and B exist, set A will be a subset of set B if all of set A's items are also present in set B.
A power set can be thought of as a container for all the subsets of a given set.
Power Set Definition
A power set is the set or group of all subsets for any given set, including the empty set indicated by {}, or, ϕ.
Example
Set A = {1,2,3}
Here,
No. of elements in the set = 3.
Now,
Let’s find the power set of set A.
Set A = {1,2,3}
Set A subset = {}, {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} }
Elements in the power set = 23 = 8
The cardinality of a Power Set
The total number of elements in a set is known as its cardinality. The list of all the subsets of a set is included in a power set.
It is to be noted that:
The number of subsets in total for a set of 'n' elements is given by 2n
The elements of a power set are the subsets of a set. Hence, the cardinality of a power set is given by:
|P(A)| = 2n
Where, n = No. of elements in the given set.
Example
Set A = {1,2} so, n = 2
Now, the number of subsets in a power set of A will be:
Subsets of A = {}, {1},{2},{1,2}
and
|P(A)| = 2n = 22 = 4.
Power Set Properties
A power set is a set that contains a list of all subsets. P(A) is a power set with n elements that have the following properties:
A set's total number of elements is \[2^{n}\].
A power set's definite element is an empty set.
There is just one element in the power set of an empty set.
A set with a finite number of elements has a finite power set.
An infinite set's power set has an infinite number of subsets.
Cantor's diagonal argument proves that the Power set of a set is much bigger than the original set.
In the Power set of A, if the number of elements is\[2^{n}\], then n is the number of elements in the original A.
For a set of natural numbers, we can map the resulting Power set, P(S), with the real numbers.
When operating with the union of sets, the intersection of sets and the complement of sets, Power set, P(S) of set S denotes the Boolean Algebra.
Power Set Generator
If the given set has n members, then the Power set will have \[2^{n}\] members.
For example
If set A has 2 natural numbers {1, 2}, then the Power set of Natural numbers will be \[2^{2}\] = 4 elements.
So, the power set will have {1}, {2}, {1, 2} and a Null set.
P(A) = {{1}, {2}, {1, 2}, {}}
Similarly, if Set B has 3 natural numbers say 1/2, 1/4, 1/6. Then the Power set of Real numbers will be \[2^{3}\] = 8 elements.
P(B)= {{1/2}, {1/4}, {1/6}, {1/2, 1/4}, {1/4 , 1/6}, {1/6, 1/2}, {1/2, 1/4, 1/6}, {}}
From this, we can conclude that the Power set will always be larger than the original set. Also, the power set elements denote the number of subsets in the Power set of a Power set.
Power Set of Null Set
This set is also called as “Power set of empty set” or “Power set of Phi (∅)”. The Power set of a Null set is Zero.
Properties of Null set:
There are zero elements in a Null set.
It is one of the subsets in the Power set.
It is represented by {} or ∅.
Problems
1. Find the Power Set and the Total Number of Elements in the Power Set if the Set A= {3, 5, 7}.
Ans: Here the set A is having a 3 set of natural numbers. Therefore the total number of elements in the Power set is \[2^{3}\] = 8.
The subsets are as follows:
{3}
{5}
{7}
{3, 5}
{5, 7}
{7, 3}
{3, 5, 7}
Null set {} or ∅
Therefore the Power set of Natural numbers will be P(A) = {{3}, {5}, {7}, {3, 5}, {5, 7}, {7, 3}, {3, 5, 7}, {}}
2. In a Cage, there are Four Sets of Animals: a Dog, Cat, Tiger and Mice. Find the Total Number of Animals in the Power Set and Also Find the Power Set of these Animals.
Ans: Here we have 4 animals, let us assign Dog as D, Cat as C, Tiger as T and Mice as M. There the set C= {D, C, T, M}. So the total number of animals in the Power set of C will be \[2^{4}\] = 16.
The subsets of the animals are as follows:
{D}
{C}
{T}
{M}
{D, C}
{D, T}
{D, M}
{C, T}
{C, M}
{T, M}
{D, C, T}
{D, C, M}
{D, T, M}
{C, T, M}
{D, C, T, M}
Null set {} or ∅
Therefore the Power set of the animals will be
P(C) = {{D}, {C}, {T}, {M}, {D, C}, {D, T}, {D, M}, {C, T}, {C, M}, {T, M}, {D, C, T}, {D, C, M}, {D, T, M}, {C, T, M}, {D, C, T, M}, {}}
3. Find the Power Set of set A = {3, 9, 11} and a total number of elements.
Solution:
A = {3,9,11}.
No. of elements of A = 3.
Total number of elements of P (A) = \[2^{n}\] = 3 = 8.
P (A) = { {}, {3}, {9}, {11}, {3,9}, {3,11}, {9,11}, {3,9,11} }
4. How many elements have P(A) if A = φ?
Solution:
Total number of elements of a null set = 0 (i.e. n = 0)
No. of elements of the power set of a null set = \[2^{n}\] = 0 = 1.
Hence, the null set itself is the only element of a power set of the null set.
Conclusion
Students will ask why we need a power set. The key thing that a power set is useful for is to have a world to take place in other things. A lot of mathematics fields begin by defining certain subsets of a set as unique. Topology deals with open sets, measure theory deals with measurable sets, a non-principal ultrafilter requires even non-standard analysis to get going. But a Power set puts all these things under a single set.
FAQs on What Are Power Sets in Maths?
1. What is a power set in Maths?
A power set is a fundamental concept in set theory. For any given set 'A', the power set of A, denoted as P(A), is the set containing all possible subsets of A. This collection includes the empty set (∅) and the set A itself.
2. How do you find the power set of a set with two elements, like A = {a, b}?
To find the power set of A = {a, b}, you must list all of its possible subsets. Following the definition, the subsets are:
- The empty set: ∅
- Subsets with one element: {a}, {b}
- The subset with two elements (the set itself): {a, b}
Therefore, the power set is P(A) = {∅, {a}, {b}, {a, b}}.
3. Can you provide an example of the power set for a set with three elements, such as B = {1, 2, 3}?
Certainly. For the set B = {1, 2, 3}, its power set, P(B), is the collection of all its 8 possible subsets. These are:
P(B) = { ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }.
This includes the empty set, all single-element subsets, all two-element subsets, and the original set itself.
4. What is the formula to calculate the number of elements in a power set?
The formula to determine the number of elements, also known as the cardinality, of a power set is 2n. Here, 'n' represents the number of elements in the original set. For example, if a set has 3 elements (n=3), its power set will have 23 = 8 elements.
5. What is the correct symbol or notation used to represent a power set?
The standard notation for the power set of a given set 'S' is P(S). It is written using a capital 'P' followed by the name of the set in parentheses. Some advanced texts may also use the notation ℘(S), using a script 'P', to denote it.
6. Why is the number of elements in a power set always a power of 2 (i.e., 2^n)?
The formula 2n arises from the choices available for each element when forming a subset. For every element in the original set (of which there are 'n'), you have exactly two choices: either to include that element in the subset or not to include it. Since there are 'n' elements and each has 2 independent choices, the total number of possible combinations (subsets) is 2 multiplied by itself 'n' times, which results in 2n.
7. What is the power set of an empty set (∅), and why isn't it also empty?
This is a common point of confusion. The power set of an empty set (∅) is P(∅) = {∅}. It is not empty because a power set must contain all possible subsets of the original set. The empty set has exactly one subset: itself (the empty set). Therefore, its power set contains this single element. A set containing one element is, by definition, not an empty set.
8. How is the concept of a power set related to subsets and the null set?
These concepts are intrinsically linked. A power set is defined as the set of all subsets of a given set. The null set (or empty set, ∅) plays a special role because it is considered a subset of every set. Consequently, the null set is always an element of every power set. Similarly, the original set itself is always a subset of itself, so it is also always an element of its own power set.
9. What is the main difference between the elements of a set and the elements of its power set?
The fundamental difference lies in their nature. The elements of an original set 'A' are the individual items themselves (e.g., numbers, letters). In contrast, the elements of its power set, P(A), are not individual items but are themselves sets. Each element within P(A) is a distinct subset of A. For instance, if A = {1, 2}, its elements are the numbers 1 and 2. The elements of P(A) are the sets ∅, {1}, {2}, and {1, 2}.
