Subset Calculator

A subset is a set where every element is also contained in another set (the superset). If A = {1, 2} and B = {1, 2, 3}, then A is a subset of B because both elements of A (1 and 2) are also in B. We write this as A ⊆ B. The empty set ∅ is considered a subset of every set, and every set is a subset of itself.

To calculate the number of subsets, use the formula 2ⁿ where n is the number of elements in the set. For example, a set with 4 elements has 2⁴ = 16 subsets. This works because each element has 2 choices: either it's included in a subset or it isn't, giving 2 × 2 × ... × 2 (n times) = 2ⁿ total combinations.

A subset (⊆) can be equal to the original set, while a proper subset (⊂) must be strictly smaller and cannot equal the original set. For set {1, 2, 3}: the subset {1, 2, 3} is an improper subset (equals the original), while {1, 2}, {1}, and ∅ are all proper subsets. A set with n elements has 2ⁿ - 1 proper subsets.

The power set P(A) is the set containing all possible subsets of A, including the empty set and A itself. To find it: (1) Start with the empty set ∅, (2) List all 1-element subsets, (3) List all 2-element subsets, and continue until (4) You include the full set. For A = {a, b}, P(A) = {∅, {a}, {b}, {a, b}}. The power set always has 2ⁿ elements.

Yes! The empty set (∅) is a subset of every set, including itself. This is a "vacuous truth" in logic — the statement "every element of ∅ is in A" is true because there are no elements in ∅ that could violate this condition. So ∅ ⊆ {1, 2, 3}, ∅ ⊆ {a, b}, and even ∅ ⊆ ∅ are all true statements.

To check if A ⊆ B, verify that every element in A is also in B. Go through each element of A one by one and check if it appears in B. If even one element of A is not found in B, then A is not a subset of B. For example, {1, 2} ⊆ {1, 2, 3} is true (both 1 and 2 are in B), but {1, 4} ⊆ {1, 2, 3} is false (4 is not in B).

The symbol ⊆ means "is a subset of" and allows the sets to be equal (A ⊆ A is true). The symbol ⊂ means "is a proper subset of" and requires A to be strictly smaller than B (A ⊂ A is false). Some textbooks use ⊂ for regular subsets, so always check your course's convention. The notation ⊊ explicitly means proper subset in all conventions.

A set with n elements has C(n, k) = n! / (k!(n-k)!) subsets of size k. This is the binomial coefficient, also called "n choose k". For example, a set with 5 elements has C(5, 2) = 10 subsets with exactly 2 elements. The sum of all C(n, k) for k = 0 to n equals 2ⁿ, the total number of subsets.

An improper subset is a subset that equals the original set. Every set has exactly one improper subset: itself. For A = {1, 2, 3}, the improper subset is {1, 2, 3}. All other subsets (∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}) are proper subsets. The number of improper subsets is always 1, regardless of set size.

Yes! This subset calculator is 100% free with no signup required. Use it unlimited times on any device — desktop, tablet, or mobile. All calculations happen instantly in your browser with no data stored on our servers. Perfect for students studying set theory, teachers creating examples, and anyone working with discrete mathematics or combinatorics!