Subset Calculator — Generate All Subsets & Count Power Set

What is a subset of a set?

Let A and B be two sets. We say that A is a subset of B if every element of A is also an element of B. In other words, A consists of some (possibly all) of the elements of B but doesn't have any elements that B doesn't have. If A is a subset of B, we can also say that B is a superset of A.

Examples:

• The empty set ∅ is a subset of any set.
• {1, 2} is a subset of {1, 2, 3, 4}.
• ∅, {1} and {1, 2} are three different subsets of {1, 2}.
• Prime numbers and odd numbers are both subsets of the set of integers.

Power set definition

The set of all possible subsets of a set (including the empty set and the set itself) is called the power set of a set. We usually denote the power set of any set A by P(A). Note that the power set consists of sets; in particular, the elements of A are NOT the elements of P(A)!

Examples:

• If A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}.
• P(∅) = {∅}.

As you can see in the examples, the power set always has more elements than the original set.

What is a proper subset?

A is a proper subset of B if A is a subset of B and A isn't equal to B. In other words, A has some but not all of the elements of B, and A doesn't have any elements that don't belong to B. We can also say that B is a proper superset of A.

Examples:

• {1} and {2} are proper subsets of {1, 2}.
• The empty set ∅ is a proper subset of {1, 2}.
• But {1, 2} is NOT a proper subset of {1, 2}.
• Prime numbers and odd numbers are two distinct proper subsets of the set of all integers.

How to use this subset calculator

1. Enter your set elements as a comma-separated list (e.g. 1, 2, 3 or a, b, c, d).
2. Optional — enter k (subset size): enter 0 to list all subsets (power set), or enter a specific number k to see only the subsets that have exactly k elements.
3. Click Calculate to see all subsets and counts instantly.

The calculator generates the full power set for sets of up to 15 elements. For larger sets it shows the counts only (since 2¹⁶ = 65,536 subsets would be impractical to list). When k is specified, subsets of that exact size are shown for any n up to 20 (provided the count doesn't exceed 500).

Example of how to find subsets and proper subsets

Let's find all subsets of the set A = {a, b, c}:

Total subsets: 2³ = 8. Proper subsets: 2³ − 1 = 7.

Number of subsets and proper subsets of a set

If a set has n elements, then:

• Total subsets = 2ⁿ (this is the size of the power set P(A)).
• Proper subsets = 2ⁿ − 1 (all subsets except the set itself).
• Subsets of size k = C(n, k) = n! / (k! × (n − k)!) (binomial coefficient).

Example of how to find the number of subsets

How many subsets does A = {1, 2, 3, 4, 5} have?

• n = 5
• Total subsets: 2⁵ = 32
• Proper subsets: 2⁵ − 1 = 31
• Subsets of size 2: C(5, 2) = 5! / (2! × 3!) = 10
• Subsets of size 3: C(5, 3) = 10

Subset vs. proper subset facts

• There is no set without a subset — each set has at least one subset: the empty set ∅.
• For each set, there is only one subset that is NOT a proper subset: the set itself.
• There is exactly one set with no proper subsets: the empty set.
• Every non-empty set has at least two subsets (itself and ∅) and at least one proper subset (∅).
• Each set has one more subset than it has proper subsets.

Notation

The symbols used for subsets vary by convention:

• ⊆ (A ⊆ B): A is a subset of B (may be equal).
• ⊊ (C ⊊ B): C is a proper subset of B (strictly smaller).
• Some textbooks use ⊂ for subsets and ⊊ for proper subsets.

To avoid ambiguity, use ⊆ for subsets and ⊊ for proper subsets — the tiny equal/unequal sign at the bottom makes the meaning explicit.

FAQs

Is the empty set a subset of every set?

Yes. The empty set ∅ is a subset of every set because there are no elements in ∅ that could violate the subset rule.

Is every set a subset of itself?

Yes, every set A satisfies A ⊆ A. This is why it is NOT a proper subset of itself.

How many subsets does an empty set have?

Exactly one: the empty set itself. So P(∅) = {∅}, and 2⁰ = 1.

What is the difference between a subset and a proper subset?

A subset can equal the original set; a proper subset must be strictly smaller (it cannot equal the original set).