Power Set

Introduction

A power set consists of all the subsets from a given set along with an empty set as well as the set itself. A power set is written as P(S). The number of elements that constitute the power set is expressed by 2n. It is a holder of the set with all its elements. In other words, when one defines a power set, it involves the groups of all the subsets formed, null/empty set, and the set itself. Therefore, while a set only puts forward its subsets, a power set puts forward a more elaborate expression of the set. This definition of power set will be clearer with examples and an understanding of its properties.

Power set meaning

To understand power set meaning one must have an overview of what sets and subsets mean. A set is a mere collection of a number of entities or objects. These objects can be finite or infinite in number. The objects that are within the brackets {} of a set are referred to as elements. Subsets are those minor sets that are within a set. Subsets can be seen as smaller parts of a whole which is the set. Therefore, the power set of a particular set, suppose Set A, will have all the elements of Set A- its subsets, empty set, the entire set. 

Example of power set

Let’s consider X the set {a,b,c}. Therefore, all the subsets of the set X are:-

{} – Empty set/null set

{a}

{b}

{c}

{a,b}

{a,c}

{b,c}

{a,b,c}

Then, the power set of X becomes:-

P(X) = {{}, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}. 

The cardinality of power set

Cardinality of a set brings forward the total number of elements in that particular set. Therefore, from a set it can be determined how many elements would be there in the power set through cardinality. The number of subsets, in total, is denoted by ‘n’. The total number of elements can be identified by applying it to 2n. The cardinality of a particular power set is put forward by:-

|P(S)| = 2n

For instance, Set X= (1,2,3)

Therefore, n=3

|P(X)| = 2n=23 =2*2*2= 8

Subsets of X are as follows:- 

{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} 

Therefore, there are 8 elements. 

Properties of power set

A power set has certain properties which make it distinct by its definition. These properties are as follows:-

• A set has 2n (2 to the power n) elements, in total
• An empty set, also called null set, is a part of the power set, too. An empty set cannot be separated as it is an element
• The empty set has only one element, in the power set. Other sets are repetitive in a power set, like {2}, {2,3}, where element 2 is repetitive
• When a set has a finite/a fixed number of elements, then the power set of that set also has a fixed number of elements
• Similarly, when a set has infinite elements, the power set of that particular set, as well as infinite elements. For instance, Set A has subsets that are multiples of two, {2,4,6,8,………}. Therefore, the power set of A will also have infinite elements. However, even if the elements do not have a fixed number, the power set definitely exists for such a set

Conclusion

What is a power set? Define power set or power set definition- all these questions have been thoroughly explained throughout. To ease the understanding of power set an elaborate example has been provided. Moreover, the cardinality of a power set has been elaborated and shown through calculations. All the formulas required while learning power sets have been mentioned and applied in the calculation. Also, the properties of power sets have been enumerated, as well. In the end, the FAQs will help to clear the core concept.