Power Sets

Introduction

Set means the collection of varied objects. For two sets, A and B, if all B elements are included in set A, then we call set B to be the subset of set A. For example, we have set A={2, 4, 6, 8, 10, 12} and set B={2,4,6}. Here we have all the elements of B included in set A. Therefore, B is the subset of A. Power set includes all the subsets of a particular set, including the empty set. In simple words, the subsets are the elements of the power set. Let us learn about the power sets, cardinality, and other properties through this power set study material. 

Understanding Power Set

As defined earlier, the power set contains all the subsets of a given set. The power set also includes a null set or empty set, which is denoted by { } or . The power set is denoted as P(S).

For example, if we have a set A={2, 4, 6} then we have possible subsets as { }, {2}, {4}, {6}, {2, 4}, {4, 6}, {2, 6}, {2, 4, 6}. 

Therefore, the power set of the given set A {2, 4, 6} is: 

P(S) = { { }, {2}, {4}, {6}, {2, 4}, {4, 6}, {2, 6}, {2, 4, 6} }

Properties of Power Set

There are various properties of the power set that define them. These properties are:

1. The cardinality of Power Set

The cardinality of the set means the total number of elements that it contains. Suppose we have a set A={2, 4, 7}, here the set has 3 elements; therefore, the cardinality of set A is 3. 

Now, the power set contains all the subsets. So, we have a total number of subsets for a given set equal to 2n, where n is the number of elements of the set. Therefore, the cardinality of the power set becomes equal to the number of possible subsets. In the above example, there are three elements in set A. Therefore, cardinality of the power set would be 23, i.e., 8.

2. Power Set of Empty Set

The null or empty set is also an element of the power set. The empty set is the set with no elements and is represented by . We know that the number of elements in the power set equals 2n, where n is the number of elements in the set. Now the empty set has zero elements; therefore, cardinality of the empty set becomes 20, i.e., 1. 

Thus, the power set of an empty set has only one element and is represented as: 

P(S) = { }. 

3. Power Set of Countable Set

The countable set is the one in which the element can be counted. It can be both a finite set and infinite set. For example, the set A={a, b, c, d, e} are countably finite. In comparison, the set B={1, 2, 3, 4, …..} is infinite. 

• If the set has a finite number of elements, then the power set would also be finite. 
• If the set has infinite elements, then the power set would also be uncountably infinite. 

4. Power Set of Uncountable Set

The uncountable set is the one whose elements cannot be counted. It is always an infinite set.

The power set of an uncountable set is always infinite and uncountable. For example, the set of real numbers is uncountable; therefore, the power set would also be infinite and uncountable. 

Power Set and Binomial Theorem

The binomial theorem and power set are related to each other. For a set with n elements, the number of combinations C (n, k) gives the number of subsets with k elements. 

For example, below, we have given the power set of a set that has 5 elements. 

• C (4,0) = We get the combination as 1; therefore, there will be only one subset. Since k = 0, the subset will have 0 elements. Thus it means 1 empty subset.   
• C (4, 1) = The combination would be 4; therefore, there will be 4 subsets with 1 element. 
• C (4, 2) = The combination would be 6; therefore, there will be 6 subsets with 2 elements. 
• C (4, 3) = There would only be 4 subsets with three elements. 
• C (4, 4) = There would be only 1 subset with four elements. 

Thus, the power set becomes the sum of all the subsets. That is: 

P(S) = 1 subset (0 element) + 4 subsets (1 element) + 6 subsets (2 elements) + 4 subsets (3 elements) + 1 subset (4 elements)

Therefore, the power set will have a total of 16 elements. 

From the above example, we can represent the relation | 2S| from the formula:

Recursive Definition of Power Set

In mathematics, the recursive definition is used to define a particular set concerning another element. The recursive definition of the power set of finite set S is given as: 

Here we understand that the power set of a null set is a singleton. It has only one element, which is the empty set itself. For a finite set S, the power set is the union of the power set of T and the power set of T, whose every element is expanded with e. 

Solved Examples

Q1. What is the cardinality of a set A {2, 4, 6, 8, 10, 12, 14}

Solution: we have the number of elements of set A equal to 7. Therefore, the total number of subsets will be 27, i.e. 128. Therefore, the cardinality of the set is 128.

Q2. Find the number of elements in the power set of a set with n+1 elements. 

Solution: The number of elements in the power set with ‘n’ element is 2n. Therefore, the number of elements in the power set with the n+1 element would be 2n+1. 

Conclusion

We get a complete understanding of the power set  with this power set study material. A power set is a set that includes all the subsets of a particular set. The power set has great importance in the set theory in mathematics, defining the universe of set theory. Together with the union, and the intersection of the set, the power set is seen as an example of Boolean Algebra. Check our website for more study material notes on power sets.