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, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between tw ...
sign with a third line. The triple bar character in
Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
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 an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well.
In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
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 the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, it is used with two different but related meanings. It can refer to the
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
connective, also called material equivalence. This is a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
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 logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
al notion of
logical equivalence
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending o ...
, according to which two formulas are logically equivalent when all
models
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
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 was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
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:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
or an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
(although not the only one; other common choices include ~ and ≈). Particularly, in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 mod ...
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 and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(who first used it with this meaning in 1801) to mean modular congruence: if ''N'' divides ''a'' − ''b''.
In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, 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 such that all directed paths in the diagram with the same start and endpoints lead to the s ...
, 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, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.a\mathbinb.
botanical nomenclature
Botanical nomenclature is the formal, scientific naming of plants. It is related to, but distinct from taxonomy. Plant taxonomy is concerned with grouping and classifying plants; botanical nomenclature then provides names for the results of this ...
, the triple bar denotes homotypic
synonyms
A synonym is a word, morpheme, or phrase that means exactly or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are ...
(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 to which the scientific name of that organism is formally attached. In other words, a type is an example that serves to anchor or centralizes th ...
), to distinguish them from heterotypic synonyms (those based on different type specimens), which are marked with an
equals sign
The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between tw ...
.
In
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
, the triple bar can be used to represent a
triple bond
A triple bond in chemistry is a chemical bond between two atoms involving six bonding electrons instead of the usual two in a covalent single bond. Triple bonds are stronger than the equivalent single bonds or double bonds, with a bond order o ...
between atoms. For example, HC≡CH is a common shorthand for
acetylene
Acetylene ( systematic name: ethyne) is the chemical compound with the formula and structure . It is a hydrocarbon and the simplest alkyne. This colorless gas is widely used as a fuel and a chemical building block. It is unstable in its pure ...
web
Web most often refers to:
* Spider web, a silken structure created by the animal
* World Wide Web or the Web, an Internet-based hypertext system
Web, WEB, or the Web may also refer to:
Computing
* WEB, a literate programming system created by ...
, 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 c ...
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 Jacob E. "Jack" Goldman, chief scientist of Xerox Corporation, the company was originally a division of Xero ...
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
Material Design
Material Design (codenamed Quantum Paper) is a design language developed by Google in 2014. Expanding on the "cards" that debuted in Google Now, Material Design uses more grid-based layouts, responsive animations and transitions, padding, and d ...
guidelines and many Android apps and web apps that follow these guidelines make use of the hamburger menu.