Triple bar

≢
≢
Not identical to
	U+2262 ≢ NOT IDENTICAL TO (&nequiv;, &NotCongruent;)

≡
≡
Identical to
	U+2261 ≡ IDENTICAL TO (&Congruent;, &equiv;)

This article contains special characters. Without proper rendering support, you may see question marks, boxes, or other symbols.

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 ⟨=⟩ with a third line.

Encoding

The triple bar character in Unicode is code point U+2261 ≡ IDENTICAL TO (&Congruent;, &equiv;).^[1] The closely related code point U+2262 ≢ NOT IDENTICAL TO (&nequiv;, &NotCongruent;) is the same symbol with a slash through it, indicating the negation of its mathematical meaning.^[1]

In LaTeX mathematical formulas, the code \equiv produces the triple bar symbol and \not\equiv produces the negated triple bar symbol $\not \equiv$ as output.^[2]

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.^[3] This is a binary operation whose value is true when its two arguments have the same value as each other.^[4] Alternatively, in some texts ⇔ is used with this meaning, while ≡ is used for the higher-level metalogical notion of logical equivalence, according to which two formulas are logically equivalent when all models give them the same value.^[5] 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.^[6]

In mathematics, the triple bar is sometimes used as a symbol of identity or an equivalence relation (although not the only one; other common choices include ~ and ≈).^[7]^[8] Particularly, in geometry, it may be used either to show that two figures are congruent or that they are identical.^[9] In number theory, it has been used beginning with 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.^[10]^[11]

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.^[12]

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.^[13] An alternative notation for this usage is to typeset the letters "def" above an ordinary equality sign, $a\mathrel {\stackrel {\scriptscriptstyle \mathrm {def} }{=}} b$ .^[14] Similarly, another alternative notation for this usage is to precede the equals sign with a colon, $a:=b$ . 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 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.^[15]

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^[16] (systematic name: ethyne).

Application design

In mobile, web, and general application 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.^[17] Usage of this symbol dates back to the early computer interfaces developed at Xerox PARC in the 1980s.^[18] 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 guidelines and many Android apps and web apps that follow these guidelines make use of the hamburger menu.

References

^ ^a ^b New Hart's Rules: The Oxford Style Guide, Oxford University Press, 2014, p. 295, ISBN 978-0-19-957002-7.
^ Lamport, Leslie (1994), LaTeX: A Document Preparation System (2nd ed.), Addison-Wesley, p. 43.
^ Salmon, Merrilee H. (1999), Introduction to the Philosophy of Science, Hackett Publishing, p. 50, ISBN 978-0-87220-450-8.
^ Hurley, Patrick (2014), A Concise Introduction to Logic (12th ed.), Cengage Learning, p. 338, ISBN 978-1-285-96556-7.
^ Dube, Rakesh; Pandey, Adesh; Gupta, Ritu (2006), Discrete Structures and Automata Theory, Alpha Science Int'l Ltd., p. 277, ISBN 978-1-84265-256-5.
^ Weiner, Joan (2013), Frege Explained, Open Court, pp. 37–38, ISBN 978-0-8126-9752-0.
^ Gallian, Joseph (2009), Contemporary Abstract Algebra (7th ed.), Cengage Learning, p. 16, ISBN 978-0-547-16509-7.
^ Lambek, J.; Scott, P.J. (1986), Introduction to higher order categorical logic, Cambridge University Press, p. ix, Remark on notation: throughout this book, we frequently, though not exclusively, use the symbol ≡ for definitional equality.
^ Cajori, Florian (2013), A History of Mathematical Notations, Dover Books on Mathematics, Courier Dover Publications, p. 418, ISBN 978-0-486-16116-7.
^ Goldstein, Catherine; Schappacher, Norbert; Schwermer, Joachim (2007), The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae, Springer, p. 21, ISBN 978-3-540-34720-0.
^ Cajori (2013), p. 34.
^ Ganz, Steven E. (2007), Encapsulation of State with Monad Transformers, Ph.D. thesis, Indiana University, p. 25, ISBN 978-0-493-91365-0.
^ Meigs, John; Olmsted, Hubbell (1956), Intermediate analysis: an introduction to the theory of functions of one real variable, Appleton-Century-Crofts, p. vi.
^ Lamport (1994), p. 50.
^ "Guidelines for authors" (PDF), Taxon, 62 (1): 211–214, 2013
^ Olmsted, John; Williams, Gregory M. (1997), Chemistry: The Molecular Science, Jones & Bartlett Learning, p. 86, ISBN 978-0-8151-8450-8
^ Peterson, Clarissa (2014), Learning Responsive Web Design: A Beginner's Guide, O'Reilly Media, pp. 338–339, ISBN 978-1-4493-6369-7.
^ Cox, Norm, "The origin of the hamburger icon", Evernote

[hart-1] New Hart's Rules: The Oxford Style Guide, Oxford University Press, 2014, p. 295, ISBN 978-0-19-957002-7.

[2] Lamport, Leslie (1994), LaTeX: A Document Preparation System (2nd ed.), Addison-Wesley, p. 43.

[3] Salmon, Merrilee H. (1999), Introduction to the Philosophy of Science, Hackett Publishing, p. 50, ISBN 978-0-87220-450-8.

[4] Hurley, Patrick (2014), A Concise Introduction to Logic (12th ed.), Cengage Learning, p. 338, ISBN 978-1-285-96556-7.

[5] Dube, Rakesh; Pandey, Adesh; Gupta, Ritu (2006), Discrete Structures and Automata Theory, Alpha Science Int'l Ltd., p. 277, ISBN 978-1-84265-256-5.

[6] Weiner, Joan (2013), Frege Explained, Open Court, pp. 37–38, ISBN 978-0-8126-9752-0.

[7] Gallian, Joseph (2009), Contemporary Abstract Algebra (7th ed.), Cengage Learning, p. 16, ISBN 978-0-547-16509-7.

[8] Lambek, J.; Scott, P.J. (1986), Introduction to higher order categorical logic, Cambridge University Press, p. ix, Remark on notation: throughout this book, we frequently, though not exclusively, use the symbol ≡ for definitional equality.

[9] Cajori, Florian (2013), A History of Mathematical Notations, Dover Books on Mathematics, Courier Dover Publications, p. 418, ISBN 978-0-486-16116-7.

[10] Goldstein, Catherine; Schappacher, Norbert; Schwermer, Joachim (2007), The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae, Springer, p. 21, ISBN 978-3-540-34720-0.

[11] Cajori (2013), p. 34.

[12] Ganz, Steven E. (2007), Encapsulation of State with Monad Transformers, Ph.D. thesis, Indiana University, p. 25, ISBN 978-0-493-91365-0.

[13] Meigs, John; Olmsted, Hubbell (1956), Intermediate analysis: an introduction to the theory of functions of one real variable, Appleton-Century-Crofts, p. vi.

[14] Lamport (1994), p. 50.

[15] "Guidelines for authors" (PDF), Taxon, 62 (1): 211–214, 2013

[16] Olmsted, John; Williams, Gregory M. (1997), Chemistry: The Molecular Science, Jones & Bartlett Learning, p. 86, ISBN 978-0-8151-8450-8

[17] Peterson, Clarissa (2014), Learning Responsive Web Design: A Beginner's Guide, O'Reilly Media, pp. 338–339, ISBN 978-1-4493-6369-7.

[18] Cox, Norm, "The origin of the hamburger icon", Evernote

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]