# 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:

We can prove the statement *A* ⊆ *B* by applying a proof technique known as the element argument^{[5]}:

The validity of this technique can be seen as a consequence of Universal generalization: the technique shows *c* ∈ *A* → *c* ∈ *B* for an arbitrarily chosen element *c*. Universal generalisation then implies ∀*x*(*x* ∈ *A* → *x* ∈ *B*), which is equivalent to *A* ⊆ *B*, as stated above.

Some authors use the symbols ⊂ and ⊃ to indicate *subset* and *superset* respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇.^{[6]} For example, for these authors, it is true of every set *A* that *A* ⊂ *A*.

Other authors prefer to use the symbols ⊂ and ⊃ to indicate *proper* (also called strict) subset and *proper* superset respectively; that is, with the same meaning and instead of the symbols, ⊊ and ⊋.^{[7]}^{[1]} This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if *x* ≤ *y*, then *x* may or may not equal *y*, but if *x* < *y*, then *x* definitely does not equal *y*, and *is* less than *y*. Similarly, using the convention that ⊂ is proper subset, if *A* ⊆ *B*, then *A* may or may not equal *B*, but if *A* ⊂ *B*, then *A* definitely does not equal *B*.