# Subset

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*.

The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

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: