Subset

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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.

Contents

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,
  • 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.
  • 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 \subsetneq and \supsetneq. 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. (This statement is vacuously true.) The empty set 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 counter-intuitive examples in which both the part and the whole are infinite, and the part has the same number of elements as the whole (see Cardinality of infinite sets).

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