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 2^{n}. 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 2^{3} = 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)| = 2^{n}

### Properties of Power Set

- Power Set is much larger compared to the Original Set.
- The number of elements in the Power Set A is 2
^{n}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

2^{3} = 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 = 2^{0}

= 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.