SUMMARY / RELATED TOPICS

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 elements of B. A and B may be equal; the relationship of one set being a subset of another is called inclusion or sometimes containment. A is a subset of B may 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, the subset relation itself is the Boolean inclusion relation. If A and B are sets and every element of A is an element of B A is a subset of B, denoted by A ⊆ B, or equivalently B is a superset of A, denoted by B ⊇ A. If A is a subset of B, but A is not equal to B A is a proper subset of B, denoted by A ⊊ B, or equivalently B is a proper superset of A, denoted by B ⊋ A. For any set S, the inclusion relation ⊆ is a partial order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B.

We may partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A. When quantified, A ⊆ B is represented as ∀x. A set A is a subset of B if and only if their intersection is equal to A. Formally: A ⊆ B ⇔ A ∩ B = A. A set A is a subset of B if and only if their union is equal to B. Formally: A ⊆ B ⇔ A ∪ B = B. A finite set A is a subset of B if and only if the cardinality of their intersection is equal to the cardinality of A. Formally: A ⊆ B ⇔ | A ∩ B | = | A |; some authors use the symbols ⊃ to indicate subset and superset respectively. 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 proper superset respectively; this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y x may or may not equal y, but if x < y x does not equal y, is less than y. Using the convention that ⊂ is proper subset, if A ⊆ B A may or may not equal B, but if A ⊂ B A does not equal B; the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true.

The set D = is a subset of E =, thus D ⊆ E is true, D ⊊ E is not true. Any set is a subset of itself, but not a proper subset; the empty set, denoted by ∅, is a subset of any given set X. It is always a proper subset of any set except itself; the set is a proper subset of The set of natural numbers is a proper subset of the set of rational numbers. These are two examples in which both the subset and the whole set are infinite, the subset has the same cardinality as the whole; the set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the latter set has a larger cardinality than the former set. Another example in an Euler diagram: Inclusion is the canonical partial order in the sense that every 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 a ≤ b if and only if ⊆. For the power set P 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 T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T. Containment order Jech, Thomas. Set Theory. Springer-Verlag. ISBN 3-540-44085-2. Media related to Subsets at Wikimedia Commons Weisstein, Eric W. "Subset". MathWorld