In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements.

When quantified, ${\displaystyle A\subseteq B}$ is represented as ${\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).}$

One can prove the statement ${\displaystyle A\subseteq B}$ by applying a proof technique known as the element argument:

Let sets A and B be given. To prove that ${\displaystyle A\subseteq B,}$

The validity of this technique can be seen as a consequence of universal generalization: the technique shows ${\displaystyle (c\in A)\Rightarrow (c\in B)}$ for an arbitrarily chosen element c. Universal generalisation then implies ${\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),}$ which is equivalent to ${\displaystyle A\subseteq B,}$ as stated above.

If A and B are sets and every element of A is also an element of B, then:

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A), then:

The empty set, written ${\displaystyle \{\}}$ or ${\displaystyle \varnothing ,}$ has no elements, and therefore is vacuously a subset of any set X.