View source for Triple bar

{{Short description|Symbol with multiple meanings}}
{{about|the symbol|the horse jump|oxer|other uses|≡ (disambiguation)}}
{{Infobox punctuation mark
|mark=≡
|name=Identical to
|unicode={{unichar|2261|IDENTICAL TO|html=}}
}}
{{Infobox punctuation mark
|mark=≢
|name=Not identical to
|unicode={{unichar|2262|NOT IDENTICAL TO|html=}}
}}
{{SpecialChars}}

The '''triple bar''' or '''tribar''', '''≡''', is a symbol with multiple, context-dependent meanings indicating equivalence of two different things. Its main uses are in mathematics and logic. It has the appearance of an [[equals sign]]&nbsp;{{angbr|1==}}  with a third line.

==Encoding==
The triple bar character in [[Unicode]] is code point {{unichar|2261|IDENTICAL TO|html=}}.<ref name="hart">{{citation|title=New Hart's Rules: The Oxford Style Guide|publisher=Oxford University Press|year=2014|isbn=978-0-19-957002-7|page=295|url=https://books.google.com/books?id=btb1AwAAQBAJ&pg=PA295}}.</ref> The closely related code point {{unichar|2262|NOT IDENTICAL TO|html=}} is the same symbol with a slash through it, indicating the negation of its mathematical meaning.<ref name="hart" /> 

In [[LaTeX]] mathematical formulas, the code <code>\equiv</code> produces the triple bar symbol and <code>\not\equiv</code> produces the negated triple bar symbol <math>\not\equiv</math> as output.<ref>{{citation|first=Leslie|last=Lamport|authorlink=Leslie Lamport|title=LaTeX: A Document Preparation System|edition=2nd|publisher=Addison-Wesley|year=1994|page=43}}.</ref>

== Uses ==

=== Mathematics and philosophy ===
In [[logic]], it is used with two different but related meanings. It can refer to the [[if and only if]] connective, also called material equivalence.<ref>{{citation|title=Introduction to the Philosophy of Science|first=Merrilee H.|last=Salmon|publisher=Hackett Publishing|year=1999|isbn=978-0-87220-450-8|page=50|url=https://books.google.com/books?id=Uq7xf0rcCQIC&pg=PA50}}.</ref> This is a [[binary operation]] whose value is true when its two arguments have the same value as each other.<ref>{{citation|title=A Concise Introduction to Logic|first=Patrick|last=Hurley|edition=12th|publisher=Cengage Learning|year=2014|isbn=978-1-285-96556-7|page=338|url=https://books.google.com/books?id=qGBQAwAAQBAJ&pg=PT338}}.</ref> Alternatively, in some texts ⇔ is used with this meaning, while ≡ is used for the higher-level [[metalogic]]al notion of [[logical equivalence]], according to which two formulas are logically equivalent when all [[model (logic)|models]] give them the same value.<ref>{{citation|title=Discrete Structures and Automata Theory|first1=Rakesh|last1=Dube|first2=Adesh|last2=Pandey|first3=Ritu|last3=Gupta|publisher=Alpha Science Int'l Ltd.|year=2006|isbn=978-1-84265-256-5|page=277|url=https://books.google.com/books?id=jd_6ynlwez0C&pg=RA7-PT41}}.</ref> [[Gottlob Frege]] used a triple bar for a more philosophical notion of identity, in which two statements (not necessarily in mathematics or formal logic) are identical if they can be freely substituted for each other without change of meaning.<ref>{{citation|title=Frege Explained|first=Joan|last=Weiner|author-link= Joan Weiner |publisher=Open Court|year=2013|isbn=978-0-8126-9752-0|pages=37–38|url=https://books.google.com/books?id=SZN6e6xEOVwC&pg=PT37}}.</ref>

In mathematics, the triple bar is sometimes used as a symbol of [[Identity (mathematics)|identity]] or an [[equivalence relation]] (although not the only one; other common choices include ~ and ≈).<ref>{{citation|title=Contemporary Abstract Algebra|first=Joseph|last=Gallian|edition=7th|publisher=Cengage Learning|year=2009|isbn=978-0-547-16509-7|page=16|url=https://books.google.com/books?id=CnH3mlOKpsMC&pg=PA16}}.</ref><ref>{{citation|publisher=[[Cambridge University Press]]|date=1986|first1=J.|last1=Lambek|first2=P.J.|last2=Scott|page=ix|title=Introduction to higher order categorical logic|quote=Remark on notation: throughout this book, we frequently, though not exclusively, use the symbol ≡ for definitional equality.}}</ref> Particularly, in [[geometry]], it may be used either to show that two figures are [[Congruence (geometry)|congruent]] or that they are identical.<ref>{{citation|title=A History of Mathematical Notations|series=Dover Books on Mathematics|first=Florian|last=Cajori|authorlink=Florian Cajori|publisher=Courier Dover Publications|year=2013|isbn=978-0-486-16116-7|page=418|url=https://books.google.com/books?id=_byqAAAAQBAJ&pg=PA418}}.</ref> In number theory, it has been used beginning with [[Carl Friedrich Gauss]] (who first used it with this meaning in 1801) to mean [[modular arithmetic|modular congruence]]: <math>a \equiv b \pmod N</math> if ''N'' divides ''a'' − ''b''.<ref>{{citation|title=The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae|first1=Catherine|last1=Goldstein|authorlink=Catherine Goldstein|first2=Norbert|last2=Schappacher|first3= Joachim|last3= Schwermer|publisher=Springer|year=2007|isbn=978-3-540-34720-0|page=21|url=https://books.google.com/books?id=IUFTcOsMTysC&pg=PA21}}.</ref><ref>{{harvtxt|Cajori|2013}}, [https://books.google.com/books?id=_byqAAAAQBAJ&pg=PA34 p.&nbsp;34].</ref>

In [[category theory]], triple bars may be used to connect objects in a [[commutative diagram]], indicating that they are actually the same object rather than being connected by an arrow of the category.<ref>{{citation|title=Encapsulation of State with Monad Transformers|first=Steven E.|last=Ganz|series=Ph.D. thesis, Indiana University|year=2007|isbn=978-0-493-91365-0|page=25|url=https://books.google.com/books?id=Mb9dShHum4gC&pg=PA25}}.</ref>

This symbol is also sometimes used in place of an equal sign for equations that define the symbol on the [[left-hand side]] of the equation, to contrast them with equations in which the terms on both sides of the equation were already defined.<ref>{{citation|title=Intermediate analysis: an introduction to the theory of functions of one real variable|first1=John|last1=Meigs|first2=Hubbell|last2=Olmsted|publisher=Appleton-Century-Crofts|year=1956|page=vi}}.</ref> An alternative notation for this usage is to typeset the letters "def" above an ordinary equality sign, <math>a\mathrel{\stackrel{\scriptscriptstyle \mathrm{def}}{=}}b</math>.<ref>{{harvtxt|Lamport|1994}}, p.&nbsp;50.</ref> Similarly, another alternative notation for this usage is to precede the equals sign with a colon, <math>a:=b</math>. The colon notation has the advantage that it reflects the inherent asymmetry in the definition of one object from already defined objects.

=== Science ===
In [[botanical nomenclature]], the triple bar denotes homotypic [[synonym (taxonomy)|synonyms]] (those based on the same [[type specimen]]), to distinguish them from heterotypic synonyms (those based on different type specimens), which are marked with an [[equals sign]].<ref>{{citation |year=2013 |title=Guidelines for authors |journal=[[Taxon (journal)|Taxon]] |volume=62 |issue=1 |pages=211–214 |url=http://www.iapt-taxon.org/files/guidelines_authors.pdf }}</ref>

In [[chemistry]], the triple bar can be used to represent a [[triple bond]] between atoms. For example, HC≡CH is a common shorthand for [[acetylene]]<ref>{{citation|title=Chemistry: The Molecular Science|first1=John|last1=Olmsted|first2=Gregory M.|last2=Williams|publisher=Jones & Bartlett Learning|year=1997|isbn=978-0-8151-8450-8|page=86|url=https://books.google.com/books?id=1vnk6J8knKkC&pg=PA86}}</ref> (systematic name: ethyne).

=== Programming ===
In the [[APL_(programming_language)|APL]] programming language, the <code>≡</code> and <code>≢</code> symbols are used to compare to two arrays for equality and inequality respectively. This is in contrast to the symbols <code>=</code> and <code>≠</code> which compares the individual elements in an order array with the elements of another array.

=== Application design ===
{{See also|Hamburger button}}In [[mobile app|mobile]], [[website|web]], and general [[Application software|application]] design, a similar symbol is sometimes used as an interface element, where it is called a [[hamburger button|hamburger icon]]. The element typically indicates that a [[Web navigation|navigation menu]] can be accessed when the element is activated; the bars of the symbol may be seen as stylized menu items, and some variations of this symbols add more bars, or bullet points to each bar, to enhance this visual similarity.<ref>{{citation|title=Learning Responsive Web Design: A Beginner's Guide|first=Clarissa|last=Peterson|publisher=O'Reilly Media|year=2014|isbn=978-1-4493-6369-7|pages=338–339|url=https://books.google.com/books?id=ULTIAwAAQBAJ&pg=PA338}}.</ref> Usage of this symbol dates back to the early computer interfaces developed at [[Xerox PARC]] in the 1980s.<ref>{{citation|last1=Cox|first1=Norm|title=The origin of the hamburger icon|url=https://www.evernote.com/shard/s207/sh/022f2237-4b4f-4096-87f2-053acd228c2d/ede2672bc3f39a1b0232f84e01ca0a83|website=Evernote}}</ref> It is also similar to the icon frequently used to indicate [[text justification|justified text alignment]]. It is an oft-used component of [[Google|Google's]] [[Material Design]] guidelines and many [[Android (operating system)|Android]] apps and web apps that follow these guidelines make use of the hamburger menu.

==References==
{{wiktionary}}
{{reflist|30em}}
{{Common logical symbols}}

[[Category:Mathematical symbols]]
[[Category:Logic symbols]]