Triple bar

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 sign with a third line. The triple bar character in 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 mathematical formulas, the code \equiv produces the triple bar symbol and \not\equiv produces the negated triple bar symbol as output. 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. This is a binary operation 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 metalogical notion of logical equivalence, according to which two formulas are logically equivalent when all models give them the same value. 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. 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 ≈). Particularly, in geometry, it may be used either to show that two figures are congruent or that they are identical. 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: if N divides a − b. In , 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. 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.