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subset
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{{redirect|Superset}}{{redirect|⊃|the logic symbol|horseshoe (symbol)|other uses|horseshoe (disambiguation)}}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 also elements of B (note that 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 also 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 meet and join are given by intersection and union.

Definitions

If A and B are sets and every element of A is also an element of B, then
* A is a subset of B, denoted by A subseteq B, or equivalently * B is a superset of A, denoted by B supseteq A.
{{anchor|proper subset}}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 a proper (or strict) subset of B, denoted by A subsetneq B, or equivalently * B is a proper (or strict) superset of A, denoted by 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) defined by A leq B iff A subseteq B. We may also partially order mathcal{P}(S) by reverse set inclusion by defining A leq B iff B subseteq A.When quantified, {{math|A ⊆ B}} is represented as {{math|∀x(x ∈ A → x ∈ B)}}.BOOK, Rosen, Kenneth H., Discrete Mathematics and Its Applications, 2012, McGraw-Hill, New York, 978-0-07-338309-5, 119, 7th,

Properties

  • A set A is a subset of B if and only if their intersection is equal to A.


Formally: A subseteq B Leftrightarrow A cap B = A.
  • A set A is a subset of B if and only if their union is equal to B.


Formally: A subseteq B Leftrightarrow A cup 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 subseteq B Leftrightarrow |A cap B| = |A|.

⊂ and ⊃ symbols

File:Venn A subset B.svg|150px|thumb|right|Euler diagramEuler diagramSome authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇.{{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Real and complex analysis | publisher=McGraw-Hill | location=New York | edition=3rd | isbn=978-0-07-054234-1 | mr=924157 | year=1987|page=6}} For example, for these authors, it is true of every set A that {{nowrap|A ⊂ A}}.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 and instead of the symbols, ⊊ and ⊋.{{Citation | title=Subsets and Proper Subsets | url=http://it.edgecombe.edu/homepage/killorant/MAT140/Module1/Subsets.pdf | access-date=2012-09-07 | archive-url=https://web.archive.org/web/20130123202559weblink | archive-date=2013-01-23 | url-status=dead }} This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and

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