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Power Set Calculator

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

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Power Set Calculator

The Power Set Calculator generates the power set of a given set. It displays all possible subsets, including the empty set and the original set, without considering 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 empty set and the set itself. It is denoted as P(X). Essentially, it lists every possible combination of elements from the original set.

Example: Let X = {a, b, c}

Subsets: {}, {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, which is also the cardinality of the power set.

Example:

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

Number of elements, n = 4

Number of elements in power set = 2⁴ = 16

Subsets:

{}
{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|. For a set with 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 its subsets of a finite set are countable.

Null Set

The null (empty) set has no elements. Its power set:

Contains only the empty set
Has zero elements in the original set
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. Steps to use it:

Input:

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

Output:

Displays all subsets of the entered set.
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. This 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, with cardinality 0.

Reference

Wikipedia: Power Set and Related Concepts

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