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{{Short description|Any one of the distinct objects that make up a set in set theory}}
{{Redirect|Element (mathematics)|the concept in Category theory|Element (category theory)}}
In [[mathematics]], an '''element''' (or '''member''') '''of a [[Set (mathematics)|set]]''' is any one of the [[Equality (mathematics)|distinct]] [[Mathematical object|objects]] that belong to that set. For example, given a set called {{mvar|A}} containing the first four positive [[integer]]s {{nowrap|(<math>A = \{1, 2, 3, 4\}</math>)}}, one could say that "3 is an element of {{mvar|A}}", expressed notationally as <math>3 \in A </math>.

==Sets==
Writing <math>A = \{1, 2, 3, 4\}</math> means that the elements of the set {{mvar|A}} are the numbers 1, 2, 3 and 4.  Sets of elements of {{mvar|A}}, for example <math>\{1, 2\}</math>, are [[subset]]s of {{mvar|A}}.

Sets can themselves be elements. For example, consider the set <math>B = \{1, 2, \{3, 4\}\}</math>. The elements of {{mvar|B}} are ''not'' 1, 2, 3, and&nbsp;4.  Rather, there are only three elements of {{mvar|B}}, namely the numbers 1 and 2, and the set <math>\{3, 4\}</math>.

The elements of a set can be anything. For example the elements of the set <math>C = \{\mathrm{\color{Red}red}, \mathrm{12}, B\}</math> are the color red, the number 12, and the set {{mvar|B}}.

In logic, a set can be defined in terms of the membership of its elements as <math>(x \in y) \leftrightarrow \forall x[P_x = y]: x \in \mathfrak D y</math>.  This basically means that there is a general [[Predicate (logic)|predication]] of x called membership that is equivalent to the statement ‘x is a member of y if and only if, for all objects x, the general predication of x is identical to y, where x is a member of the [[Domain of a function|domain]] of y.’  The expression x ∈ 𝔇y makes this definition well-defined by ensuring that x is a bound variable in its predication of membership in y.  

In this case, the domain of Px, which is the set containing all dependent logical values x that satisfy the stated conditions for membership in y, is called the Universe (U) of y.  The range of Px, which is the set of all possible dependent set variables y resulting from satisfaction of the conditions of membership for x, is the [[power set]] of U such that the [[binary relation]] of the membership of x in y is any subset of the [[cartesian product]] U × 𝒫(U) (the Cartesian Product of set U with the Power Set of U).

==Notation and terminology==
The [[binary relation]] "is an element of", also called '''set membership''', is denoted by the symbol&nbsp;"∈". Writing

:<math>x \in A </math>

means that "''x'' is an element of&nbsp;''A''".<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Element|url=https://mathworld.wolfram.com/Element.html|access-date=2020-08-10|website=mathworld.wolfram.com|language=en}}</ref> Equivalent expressions are "''x'' is a member of&nbsp;''A''", "''x'' belongs to&nbsp;''A''", "''x'' is in&nbsp;''A''" and "''x'' lies in&nbsp;''A''". The expressions "''A'' includes ''x''" and "''A'' contains ''x''" are also used to mean set membership, although some authors use them to mean instead "''x'' is a [[subset]] of&nbsp;''A''".<ref name="schech">{{cite book |author = Eric Schechter |author-link = Eric Schechter |title= Handbook of Analysis and Its Foundations |publisher= [[Academic Press]] |year= 1997|isbn= 0-12-622760-8 }} p. 12</ref> Logician [[George Boolos]] strongly urged that "contains" be used for membership only, and "includes" for the subset relation only.<ref name="boolos">{{cite speech |title=24.243 Classical Set Theory (lecture) |author=George Boolos |author-link=George Boolos |date=February 4, 1992 |location=[[Massachusetts Institute of Technology]] }}</ref>

For the relation ∈ , the [[converse relation]] ∈<sup>T</sup> may be written

:<math>A \ni x</math>

meaning "''A'' contains or includes ''x''".

The [[negation]] of set membership is denoted by the symbol&nbsp;"∉". Writing
:<math>x \notin A</math>

means that "''x'' is not an element of&nbsp;''A''".

The symbol ∈ was first used by [[Giuseppe Peano]], in his 1889 work {{lang|la|[[Arithmetices principia, nova methodo exposita]]|italic=yes}}.<ref name=ken>{{cite journal|last=Kennedy|first=H. C.|date=July 1973|doi=10.1305/ndjfl/1093891001|issue=3|journal=Notre Dame Journal of Formal Logic|mr=0319684|pages=367–372|publisher=Duke University Press|title=What Russell learned from Peano|volume=14|doi-access=free}}</ref> Here he wrote on page X:

<blockquote>{{lang|la|Signum {{noitalic|∈}} significat est. Ita {{math|a {{noitalic|∈}} b}} legitur a est quoddam b; …}}</blockquote>

which means

<blockquote>The symbol ∈ means ''is''. So {{math|''a'' ∈ ''b''}} is read as a ''is a certain'' b; …</blockquote>

The symbol itself is a stylized lowercase Greek letter [[epsilon]] ("ϵ"), the first letter of the word {{wikt-lang|grc|ἐστί}}, which means "is".<ref name=ken/>

{{charmap
|2208 |name1=Element of                 |ref1char1=\in               |ref2char1=\[Element]
|2209 |name2=Not an element of          |ref1char2=\notin            |ref2char2=\[NotElement]
|220b |name3=Contains as member         |ref1char3=\ni               |ref2char3=\[ReverseElement]
|220c |name4=Does not contain as member |ref1char4=\not\ni or \notni |ref2char4=\[NotReverseElement]
|namedref1=[[LaTeX]]
|namedref2=[[Wolfram Mathematica (software)|Wolfram Mathematica]]
}}

==Examples==
Using the sets defined above, namely ''A'' = {1, 2, 3, 4}, ''B'' = {1, 2, {3, 4}} and ''C'' = {red, 12, ''B''}, the following statements are true:
*{{math|1=2 ∈ ''A''}}
*{{math|1=5 ∉ ''A''}}

*{{math|1={{mset|3, 4}} ∈ ''B''}}
*{{math|1=3 ∉ ''B''}}
*{{math|1=4 ∉ ''B''}}
*{{math|1=yellow ∉ ''C''}}

==Cardinality of sets==
{{main|Cardinality}}
The number of elements in a particular set is a property known as [[cardinality]]; informally, this is the size of a set.<ref>{{Cite web|title=Sets - Elements {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/sets-elements/|access-date=2020-08-10|website=brilliant.org|language=en-us}}</ref> In the above examples, the cardinality of the set&nbsp;''A'' is&nbsp;4, while the cardinality of set ''B'' and set ''C'' are both&nbsp;3. An infinite set is a set with an infinite number of elements, while a [[finite set]] is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers {{math|{{mset|1, 2, 3, 4, ...}}}}.

==Formal relation==
As a [[relation (mathematics)|relation]], set membership must have a domain and a range. Conventionally the domain is called the [[universe (mathematics)|universe]] denoted ''U''. The range is the set of [[subset]]s of ''U'' called the [[power set]] of ''U'' and denoted P(''U''). Thus the relation <math>\in</math> is a subset of {{math|''U'' &times; P(''U'')}}. The converse relation <math>\ni</math> is a subset of {{math|P(''U'') &times; ''U''}}.

== See also ==

* [[Identity element]]
* [[Singleton (mathematics)]]

==References==
{{Reflist}}

==Further reading==
*{{Citation|last=Halmos|first=Paul R.|author-link=Paul R. Halmos|orig-year=1960|year=1974|title=Naive Set Theory|publisher=Springer-Verlag|location=NY|edition=Hardcover|series=[[Undergraduate Texts in Mathematics]]|isbn=0-387-90092-6|url-access=registration|url=https://archive.org/details/naivesettheory0000halm_r4g0}} - "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).
*{{Citation|last=Jech|first=Thomas|author-link=Thomas Jech|year=2002|title=Stanford Encyclopedia of Philosophy|chapter=Set Theory|publisher=Metaphysics Research Lab, Stanford University|chapter-url=http://plato.stanford.edu/entries/set-theory/}}
*{{Citation|last=Suppes|first=Patrick|author-link=Patrick Suppes|orig-year=1960|year=1972|title=Axiomatic Set Theory|publisher=Dover Publications, Inc.|location=NY|isbn=0-486-61630-4|url-access=registration|url=https://archive.org/details/axiomaticsettheo00supp_0}} - Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thorough understanding of "set element".

{{Mathematical logic}}
{{Set theory}}

[[Category:Basic concepts in set theory]]