In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained in B. That is, all elements of A are also elements of B. A and B may 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 A, or A is included 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, then
For any set S, the inclusion relation ⊆ is a partial order on the set of all subsets of S defined by. We may also partially order by reverse set inclusion by defining
When quantified, is represented as.


Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇. For example, for these authors, it is true of every set A that.
Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and proper superset respectively; that is, with the same meaning and instead of the symbols, ⊊ and ⊋. This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if then x may or may not equal y, but if, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if, then A may or may not equal B, but if, then A definitely does not equal B.


Another example in an Euler diagram:

Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then ab if and only if ⊆ .
For the power set of a set S, the inclusion partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 < 1. This can be illustrated by enumerating S = and associating with each subset TS the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T.