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*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**.

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

## Definitions

If*A*and

*B*are sets and every element of

*A*is also an element of

*B*, then:

*

{{anchor|proper subset}}If *A*is a**subset**of*B*, denoted by A subseteq B, or equivalently **B*is a**superset**of*A*, denoted by B supseteq A.*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

*

For any set *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.*S*, the inclusion relation âŠ† 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 âŠ† B}} is represented as: {{math|âˆ€x{xâˆˆA â†’ xâˆˆB}}}.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|.

## âŠ‚ and âŠƒ symbols

File:Venn A subset B.svg|150px|thumb|right|Euler diagramEuler diagramSome authors use the symbols âŠ‚ and âŠƒ to indicate*subset*and

*superset*respectively; that is, with the same meaning and instead of the symbols, âŠ† and âŠ‡.{{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}} So for example, for these authors, it is true of every set

*A*that {{nowrap|

*A*âŠ‚

*A*}}.Other authors prefer to use the symbols âŠ‚ and âŠƒ to indicate

*proper*(also called strict) subset and

*proper*superset respectively; that is, with the same meaning and instead of the symbols, âŠŠ and âŠ‹.{{Citation | title=Subsets and Proper Subsets | url=http://it.edgecombe.edu/homepage/killorant/MAT140/Module1/Subsets.pdf | accessdate=2012-09-07}}{{dead link|date=February 2019}} This usage makes âŠ† and âŠ‚ analogous to the inequality symbols â‰¤ and

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