Editing Element (mathematics) (section)

==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/>

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