Power Sets

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:

1. A set's total number of elements is \[2^{n}\].

2. A power set's definite element is an empty set.

3. There is just one element in the power set of an empty set.

4. A set with a finite number of elements has a finite power set.

5. An infinite set's power set has an infinite number of subsets. 

6. Cantor's diagonal argument proves that the Power set of a set is much bigger than the original set.

7. 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.

8. For a set of natural numbers, we can map the resulting Power set, P(S), with the real numbers. 

9. 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.