# Subset Explained

In mathematics, especially in set theory, a set *A* is a **subset** of a set *B* if *A* is "contained" inside *B*. Notice that *A* and *B* may coincide. The relationship of one set being a subset of another is called **inclusion**.

## Definitions

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

*A* is a subset of (or is included in) *B*, denoted by

,

or equivalently

*B* is a **superset** of (or includes) *A*, denoted by

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

*A* is also a **proper** (or **strict**) subset of *B*; this is written as

or equivalently

*B* is a proper superset of *A*; this is written as

For any set *S*, the inclusion relation ⊆ is a partial order on the set 2^{S} of all subsets of *S* (the power set of *S*).

## The symbols ⊂ and ⊃

Some authors use the symbols ⊂ and ⊃ to indicate "subset" and "superset" respectively, instead of the symbols ⊆ and ⊇, but with the same meaning. So 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* subset and superset, respectively, in place of

and

This usage makes ⊆ and ⊂ analogous to ≤ and <. For example, if

*x* ≤

*y* then

*x* may be equal to

*y*, or maybe not, but if

*x* <

*y*, then

*x* definitely does not equal

*y*, but is strictly less than

*y*. Similarly, using the "⊂ means proper subset" convention, if

*A* ⊆

*B*, then

*A* may or may not be equal to

*B*, but if

*A* ⊂

*B*, then

*A* is definitely not equal to

*B*.

## Examples