Subset

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"Superset" redirects here. For supersets in strength training, see Strength training#Combined sets.

Euler diagram showing
A is a proper subset of B and conversely B is a proper superset of A

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B.

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 $A \subseteq B$ ,

or equivalently

B is a superset of (or includes) A, denoted by $B \supseteq A.$

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 $A\subsetneq B.$

or equivalently

B is a proper superset of A; this is written as $B\supsetneq A.$

For any set S, the inclusion relation ⊆ is a partial order on the set $\mathcal{P}(S)$ 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 the inequality symbols ≤ 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

The set {1, 2} is a proper subset of {1, 2, 3}.
Any set is a subset of itself, but not a proper subset.
The empty set, denoted by ∅, is also a subset of any given set X. It also is always a proper subset, except of itself.
The set {x: x is a prime number greater than 2000} is a proper subset of {x: x is an odd number greater than 1000}
The set of natural numbers is a proper subset of the set of rational numbers and the set of points in a line segment is a proper subset of the set of points in a line. These are examples in which both the part and the whole are infinite, and the part has the same number of elements as the whole; such cases can tax intuition.

Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, $\preceq$ ) 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 n of all ordinals less than or equal to n, then a ≤ b if and only if a ⊆ b.

For the power set $\mathcal{P}(S)$ of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product of k = |S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating S = {s₁, s₂, …, s_k} and associating with each subset T ⊆ S (which is to say with each element of 2^S) the k-tuple from {0,1}^k of which the ith coordinate is 1 if and only if s_i is a member of T.

References

Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.