Subset Calculator

What Is a Subset?

A set $A$ is a subset of a set $B$ if every element of $A$ is also an element of $B$. We write $A \subseteq B$ when $A$ may equal $B$, and $A \subsetneq B$ (or $A \subset B$) when $A$ is a proper subset, meaning $A \subseteq B$ but $A \neq B$.

The empty set $\emptyset$ is a subset of every set. Every set is also a subset of itself, but not a proper subset of itself.

How Many Subsets Does a Set Have?

If a finite set $S$ has $n$ elements, the total number of subsets is:

$$| \text{subsets of } S | = 2^n$$

Each element is independently either included or excluded, giving two choices per element and $2^n$ combinations overall.

The number of proper subsets excludes the set itself:

$$| \text{proper subsets} | = 2^n - 1$$

Subsets by Size and Binomial Coefficients

The number of subsets of $S$ with exactly $k$ elements is the binomial coefficient:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Summing over all sizes from $0$ to $n$ reproduces $2^n$ via the binomial theorem. For example, a 4-element set has subset counts $1, 4, 6, 4, 1$ by size.

Related tools: Power Set Calculator, Binomial Coefficient Calculator, and Combinations Permutations Calculator.