Power Set Generator - Find All Subsets Online

Generate the complete power set of any set with our free online power set generator. Find all 2ⁿ subsets including the empty set and the set itself.

Set (comma or space separated)

Examples:

Understanding Power Sets

The power set of a set S is the collection of ALL possible subsets of S - every combination of elements you can make, from the empty set (no elements) to the full set itself. It's called the "power" set because if your original set has n elements, the power set has 2ⁿ elements.

Think of it this way: for each element in your set, you have two choices - include it in a subset or don't. With n elements, that's 2 × 2 × 2 × ... (n times) = 2ⁿ possible combinations.

Power Set Properties

Size Formula

If |S| = n, then |P(S)| = 2ⁿ

Set with 1 element → 2¹ = 2 subsets

Set with 2 elements → 2² = 4 subsets

Set with 3 elements → 2³ = 8 subsets

Set with 10 elements → 2¹⁰ = 1,024 subsets

Key Properties

• The empty set ∅ is always in P(S)
• The set S itself is always in P(S)
• Every element of S appears in exactly half the subsets
• P(∅) = {∅} (one subset: the empty set)

Worked Examples

Example 1: Power Set of {1, 2}

Original set: S = {1, 2}

Number of elements: n = 2

Expected subsets: 2² = 4

Power set P(S) = { ∅, {1}, {2}, {1,2} }

Breakdown: 1 empty + 2 singletons + 1 full set = 4

Example 2: Power Set of {a, b, c}

Original set: S = {a, b, c}

Number of elements: n = 3

Expected subsets: 2³ = 8

Power set P(S) = {

∅,

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

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

{a,b,c}

}

Breakdown: 1 + 3 + 3 + 1 = 8 subsets

Example 3: Power Set of {red, blue}

Using color names as elements:

S = {red, blue}

P(S) = { ∅, {red}, {blue}, {red,blue} }

Elements can be anything: numbers, letters, words, objects...

Example 4: Real-World Application

A pizza shop offers 4 toppings: pepperoni, mushrooms, onions, peppers. How many different pizzas can you order (including plain cheese)?

This is the power set of {pepperoni, mushrooms, onions, peppers}

Number of possible pizzas = 2⁴ = 16

Breakdown: 1 plain + 4 one-topping + 6 two-topping + 4 three-topping + 1 four-topping = 16

You can order 16 different pizza combinations!

Quick Fact

Georg Cantor, the father of set theory, proved that the power set of any set is always strictly larger than the original set - even for infinite sets! This means there are different "sizes" of infinity. The power set of the natural numbers has the same size as the real numbers. This revolutionary idea was so controversial that some mathematicians attacked Cantor personally, contributing to his mental health struggles.

Frequently Asked Questions

Why is it called a "power" set?

The name comes from the fact that if a set has n elements, its power set has 2ⁿ (2 to the power of n) elements. The notation P(S) or ℘(S) comes from the German "Potenzmenge" meaning "power set."

What is the power set of the empty set?

P(∅) = {∅}. The empty set has one subset: itself. So |P(∅)| = 2⁰ = 1. This might seem paradoxical, but the empty set is a subset of every set, including itself.

How do I count subsets of a specific size?

Use binomial coefficients (combinations). The number of k-element subsets of an n-element set is C(n,k) = n!/(k!(n-k)!). For a 5-element set: C(5,0)=1 empty, C(5,1)=5 singletons, C(5,2)=10 pairs, C(5,3)=10 triples, C(5,4)=5 quadruples, C(5,5)=1 full set. Total: 1+5+10+10+5+1=32=2⁵.

Can a set contain itself as an element?

In standard set theory (ZFC), no - this leads to Russell's Paradox. However, the power set P(S) contains S as a subset (element of the power set), which is different. S ∈ P(S) but S ⊄ S.

What's the difference between a subset and an element?

An element is a member of a set (1 ∈ {1,2,3}). A subset is a set whose elements are all in another set ({1,2} ⊆ {1,2,3}). In the power set, the elements ARE subsets of the original set.

Where are power sets used in computer science?

Power sets appear in combinatorial optimization, feature selection in machine learning (which features to include), test case generation (which tests to run), access control (which permissions to grant), and anywhere you need to enumerate all possible combinations.

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