Power Set

Power Set is a set that includes all the Subsets along with Empty Set and the Original Set itself. To know more about Sets check out Set Theory and get a good grip on the concept. Check out Power Set Notation, Properties, How to Calculate Power Set, Power Set of Empty Set in the coming modules. For a better understanding of the Concept Power Set we even provided solved examples.

Example:

If set A = {u,v,w} is a set, then all its subsets {u}, {v}, {w}, {u,v}, {v,w}, {u,w}, {u,v,w} and {} are the elements of powerset, such as Power set of A, P(A) = {u}, {v}, {w}, {u,v}, {v,w}, {u,w}, {u,v,w} and {}

Where P(A) represents the Powerset.

Definition of Power Set

Power Set of A is defined as all the subsets within the Set A along with the Null Set and the Set itself. It is represented by P(A) and is a combination of the null set, set itself, and subsets.

How to Calculate Power Set?

If a Set has n elements then the Power Set can be obtained using the Formula 2ⁿ. It even denotes the Cardinality of a Power Set.

Example

Let us assume Set A = { x, y, z }

Number of elements: 3

Therefore, the subsets of the set are:

{ } which is the null or the empty set

{ x } { y } { z } { x, y } { y, z} { z, x } { x, y, z }

The power set P(A) = { { } , { x } { y } { z } { x, y } { y, z} { z, x } { x, y, z } }

Now, the Power Set has 2³ = 8 elements.

Power Set Notation

The number of elements of a power set is given by |A|, If A has n elements then it can be represented as |P(A)| = 2ⁿ

Properties of Power Set

• Power Set is much larger compared to the Original Set.
• The number of elements in the Power Set A is 2ⁿ where n represents the number of elements in Set A.
• Power Set of Finite Set if Finite and Countable.
• For a Set of Natural Numbers, we can do one-one mapping of the resultant set, P(S) with real numbers.
• P(S) of Set S if performed Operations like Union, Intersection, Complement denotes the Boolean Algebra.

Power Set of Empty Set

In general, Empty Set has no elements and the Power Set of Empty Set denotes the following

• A Set containing Null or Void Set.
• Empty Set is the only Subset.
• It Contains No Elements in the Set.

Solved Examples on Power Set

1. Find the Power Set of Z = {4,7,8}?

Solution:

Given Set Z= {4,7,8}

Number of Elements n = 3

2³ = 8, that shows there will be eight elements of power set of Z

Subsets of Z are {{} {4}, {7}, {8}, {4, 7} {7, 8} {4, 8} {4, 7, 8}}

2. What is the Power set of an Empty set?

Solution:

Number of Elements in Empty Set = 0

No. of Elements in Power Set = 2⁰

= 1

Thus, there is only one element in the power set and that is an empty set.

Power Set of Empty Set P(E) = 1.