Subset

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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, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment.

The subset relation defines a partial order on sets.

The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

Definitions[edit]

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 which is not an element of 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 ⊃[edit]

Some authors[1]:p.6 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[2] 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, and 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[edit]

  • 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 is also always a proper subset of any set except itself.
  • The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}
  • 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 cardinality (number of elements) as the whole; such cases can tax one's intuition.

Other properties of inclusion[edit]

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 ab 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, …, sk} and associating with each subset TS (which is to say with each element of 2S) the k-tuple from {0,1}k of which the ith coordinate is 1 if and only if si is a member of T.

See also[edit]

References[edit]

  1. ^ Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 924157 
  2. ^ Subsets and Proper Subsets, retrieved 2012-09-07 

External links[edit]