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.
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:
The empty set, written or is a subset of any set X and a proper subset of any set except itself, the inclusion relation is a partial order on the set (the power set of S—the set of all subsets of S^{[1]}) defined by . We may also partially order by reverse set inclusion by defining
When quantified, is represented as ^{[2]}
We can prove the statement by applying a proof technique known as the element argument^{[3]}:
Let sets A and B be given. To prove that
- suppose that a is a particular but arbitrarily chosen element of A
- show that a is an element of B.
The validity of this technique can be seen as a consequence of universal generalization: the technique shows for an arbitrarily chosen element c. Universal generalisation then implies which is equivalent to as stated above.
The set of all subsets of is called its powerset, and is denoted by . The set of all -subsets of is denoted by , in analogue with the notation for binomial coefficients, which count the number of -subsets of an -element set. In set theory, the notation is also common, especially when is a transfinite cardinal number.
Some authors use the symbols and to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols and ^{[4]} For example, for these authors, it is true of every set A that (a reflexive relation).
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 as and instead of the symbols and ^{[5]} 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 (a irreflexive relation). 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:
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 if and only if
For the power set of a set S, the inclusion partial order is—up to an order isomorphism—the Cartesian product of (the cardinality of S) copies of the partial order on for which This can be illustrated by enumerating , and associating with each subset (i.e., each element of ) the k-tuple from of which the ith coordinate is 1 if and only if is a member of T.