Power Set Calculator

Power Set Calculator

Enter a set of numbers, and the calculator will determine the power set, cardinality, subsets, and proper subsets with detailed calculations.

Power Set Calculator:

This Power Set Calculator generates the power set of a given set. It shows all possible subsets, including the empty set and the original set, without concern for the order of elements.

What is a Power Set?

In mathematics, a power set is the set of all subsets of a given set, including the null (empty) set and the set itself. It is denoted as P(X). Simply put, it lists all possible combinations of elements from the original set.

For example, if X = {a, b, c}, the subsets are:

{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}

Thus, the power set is:

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

Power Set Calculator

For interval-related calculations, you can also use an Interval Notation Calculator.

How to Find Power Sets?

If a set has n elements, its power set contains 2ⁿ elements. This also gives the cardinality of the power set.

Example:

Let X = {1, 2, 3, 4}

n = number of elements = 4

Number of elements in the power set = 2⁴ = 16

Subsets are:

{}
{1}, {2}, {3}, {4}
{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}
{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}
{1, 2, 3, 4}

Notation:

The number of elements in a set X is written as |X|. If X has n elements:

|P(X)| = 2ⁿ

Properties of Power Sets:

The power set has more elements than the original set.
The number of elements in P(X) is 2ⁿ, where n = |X|.
The set and subsets of a finite set are countable.

Null Set:

The null set has no elements. Its power set is:

A set containing only the empty set
It has zero elements
The null set is the only subset of itself

How the Power Set Calculator Works?

The power set generator quickly creates all possible subsets of a given set. Follow these steps:

Input:

Enter the elements of the set, separated by commas.
Click the calculate button to generate all subsets.

Output:

The calculator displays all subsets of the entered set.
It shows the total number of possible subsets.

FAQ:

Can a power set be null or empty?

The power set must contain at least the empty set. For a null set, |P({})| = 2⁰ = 1. It is the smallest power set and a proper subset of all other power sets.

Is a null set finite or infinite?

The null set is finite, and its cardinality is 0.

Reference:

Wikipedia: Power Set and Related Concepts