How to Form the Power Set and Explain their Governing Rules

Power Set Introduction

The power set is defined as the set of all subsets for any given set, including the empty set, which is denoted by {}, or, ϕ. A set that has ‘n’ elements can have a total of 2nsubsets. For instance, let Set A = {1,2,3}, therefore, the total number of elements in set A is 3. Therefore, there will be 23elements 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 Power Set

The cardinality of a set means the total number of elements in a given set. The total number of subsets for a set of ‘n’ elements is 2n. Since the subsets of a set are the elements of a power set, the cardinality of a power set is

 |P(A)| = 2n. So, n = total number of elements in any given set.

Power Set Properties

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

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

Power Set Proof

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

The proof of the power set follows the pattern of the mathematical induction. For starters, let us consider the case of a set with no elements, in other words an empty set.

Case 1: The given set with no elements. Let A = {}.
Here, the power set of A 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 null set also, by the formula of the cardinality of power set, there will be 2n  number of element in the power sets, which are equal to 20or 1.

Case 2:
This is an inductive step since it is to be proved that P(n) → P(n+1). This means, if a set that has n number of elements has 2nsubsets, that means a set that has ‘n+1’ elements will have 2n+1 subsets.
To prove this, let us take two sets ‘X’ and ‘Y’ with the following elements.
X = {a1, a2, a3,a4, an} and
Y = {a1, a2, a3, a4, an, an+1}

The cardinality of the two sets ‘X’ and ‘Y’ are,
|X| = n, which means that there are 2n subsets for the set ‘X’.
|Y| = n+1
We can write Y = X U {an+1a}, this means that every subset of set ‘X’ is also a subset of set ‘Y’.
The subset of set Y may or may not contain the element an+1.
If an element of set ‘Y’ does not contain the element an+1, then it is clear that it is also an element of set ‘X’.
Also, if the subset of ‘Y’ has the element an+1, this means that the element an+1 is included in any of the 2nsubsets of the set ‘X’, So we can conclude that, set ‘Y’ has 2n subsets with the element an+1 addition to it. Therefore, set Y has both 2n subsets with element an+1 and 2n subsets without element an+1.

Conclusion

Here in this article, we have talked about the set, power set, and many more types of sets. We have explained the set with an example to understand the topic better. We have even talked about the properties of power sets.