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The triple bar, or tribar ≡, is a symbol with multiple, context-dependent meanings. It has the appearance of an
equals sign The equals sign (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar an ...

equals sign
  sign with a third line. The triple bar character in
Unicode Unicode, formally the Unicode Standard, is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's wri ...

Unicode
is code point .. The closely related code point is the same symbol with a slash through it, indicating the negation of its mathematical meaning. In
LaTeX Latex is a stable dispersion (emulsion An emulsion is a mixture of two or more liquids that are normally Miscibility, immiscible (unmixable or unblendable) owing to liquid-liquid phase separation. Emulsions are part of a more general class o ...

LaTeX
mathematical formulas, the code \equiv produces the triple bar symbol and \not\equiv produces the negated triple bar symbol as output.


Uses


Mathematics and philosophy

In
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
, it is used with two different but related meanings. It can refer to the
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
connective, also called material equivalence. This is a
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
whose value is true when its two arguments have the same value as each other. Alternatively, in some texts ⇔ is used with this meaning, while ≡ is used for the higher-level
metalogic Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how formal system, logical systems can be used to construct Validity (logic), valid and soundness, sound arguments, metalogic studies the properties of logical systems.Har ...
al notion of
logical equivalence In logic and mathematics, statements p and q are said to be logically equivalent if they are provable from each other under a set of axioms, or have the same truth value in every model (logic), model. The logical equivalence of p and q is sometimes ...
, according to which two formulas are logically equivalent when all models give them the same value.
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...
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. In mathematics, the triple bar is sometimes used as a symbol of
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
or an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
(although not the only one; other common choices include ~ and ≈). Particularly, in
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, it may be used either to show that two figures are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
or that they are identical. In number theory, it has been used beginning with
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Carl Friedrich Gauss
(who first used it with this meaning in 1801) to mean modular congruence: a \equiv b \pmod N if ''N'' divides ''a'' − ''b''. It is also used for "identical equality" of functions; one writes f \equiv g for two functions ''f'', ''g'' if we have f(x) = g(x) for all ''x''. In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, triple bars may be used to connect objects in a
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a Diagram (category theory), diagram such that all directed paths in the diagram with the same start an ...

commutative diagram
, indicating that they are actually the same object rather than being connected by an arrow of the category. This symbol is also sometimes used in place of an equal sign for equations that define the symbol on the
left-hand side In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of the equation, to contrast them with equations in which the terms on both sides of the equation were already defined. An alternative notation for this usage is to typeset the letters "def" above an ordinary equality sign, a\mathbinb.


Science

In
botanical nomenclature Botanical nomenclature is the formal, scientific naming of plants. It is related to, but distinct from taxonomy Taxonomy (general) is the practice and science of classification of things or concepts, including the principles that underlie such ...
, the triple bar denotes homotypic
synonyms A synonym is a word, morpheme A morpheme is the smallest meaningful lexical item in a language. A morpheme is not a word. The difference between a morpheme and a word is that a morpheme bound and free morphemes, sometimes does not stand alone ...
(those based on the same
type specimen In biology, a type is a particular specimen (or in some cases a group of specimens) of an organism In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Bi ...
), to distinguish them from heterotypic synonyms (those based on different type specimens), which are marked with an
equals sign The equals sign (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar an ...

equals sign
. In
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

chemistry
, the triple bar can be used to represent a
triple bond A triple bond in chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they un ...

triple bond
between atoms. For example, HC≡CH is a common shorthand for
acetylene Acetylene (systematic nameA systematic name is a name given in a systematic way to one unique group, organism, object or chemical substance A chemical substance is a form of matter In classical physics and general chemistry, matter is ...

acetylene
(systematic name: ethyne).


Application design

In mobile,
web Web most often refers to: * Spider web A spider web, spiderweb, spider's web, or cobweb (from the archaic word '' coppe'', meaning "spider") is a structure created by a spider Spiders ( order Araneae) are air-breathing arthropod An ar ...

web
, and general
application Application may refer to: Mathematics and computing * Application software, computer software designed to help the user to perform specific tasks ** Application layer, an abstraction layer that specifies protocols and interface methods used in a co ...
design, a similar symbol is sometimes used as an interface element, where it is called a hamburger icon. The element typically indicates that a 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. Usage of this symbol dates back to the early computer interfaces developed at
Xerox PARC PARC (Palo Alto Research Center; formerly Xerox PARC) is a research and development company in Palo Alto, California. Founded in 1969 by Jack Goldman, Jacob E. "Jack" Goldman, Xerox Corporation's chief scientist, the company was originally a divis ...
in the 1980s. It is also similar to the icon frequently used to indicate justified text alignment. It is an oft-used component of
Google's Google LLC is an American multinational technology company "Technology", in this context, has come to mean primarily electronics-based technology. This can include, for example, business relating to digital electronics Digital electroni ...

Google's
Material Design #REDIRECT Material Design Material Design (codenamed Quantum Paper) is a design language A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process, or the result of that ...

Material Design
guidelines and many
Android Android may refer to: Science and technology * Android (robot), a humanoid robot or synthetic organism designed to imitate a human * Android (operating system), Google's mobile operating system ** Android (operating system)#Mascot, Unnamed Androi ...

Android
apps and web apps that follow these guidelines make use of the hamburger menu.


References

{{Common logical symbols Mathematical symbols Logic symbols