Triple Bar

The triple bar, ≡, is a symbol with multiple, context-dependent meanings. It has the appearance of a "=" sign with a third line.

In logic, it has a similar meaning to the if and only if connective, ⇔. However, in some texts ⇔ is used as a symbol in logic formulas, while ≡ is for reasoning about those formulas (as in metalogic).

In mathematics it sometimes used a symbol for congruence (although not the only one). Particularly, in number theory, it has the meaning of modular congruence: if N divides a − b.

This symbol is also used when it appears in an equation which is a definition of its left-hand side, that is an equation which is not derived but instead defined. Relatedly, it is used to signify an "identity" – i.e. an equality that remains true regardless of the values of any variables that appear within.

It is also used for "identical equality" of functions; one writes for two functions f, g if we have for all x.

In chemistry, the triple bar can be used to represent a triple bond between atoms. For example, HC≡CH is a common short-hand for acetylene.

The triple bar character in Unicode as codepoint U+2261 ≡ identical to (HTML: ≡ &equiv;). LaTeX \equiv corresponds to the triple bar.

Logic

Overview

Academic areas	Argumentation theory Axiology Critical thinking Computability theory Formal semantics History of logic Informal logic Logic in computer science Mathematical logic Mathematics Metalogic Metamathematics Model theory Philosophical logic Philosophy Philosophy of logic Philosophy of mathematics Proof theory Set theory

Foundational concepts	Abduction Analytic truth Antinomy A priori Deduction Definition Description Induction Inference Logical form Logical consequence Logical truth Name Necessity Meaning Paradox Possible world Presupposition Probability Reason Reasoning Reference Semantics Statement Strict implication Substitution Syntax Truth Truth value Validity

Philosophical logic

Critical thinking and Informal logic	Analysis Ambiguity Argument Belief Bias Credibility Evidence Explanation Explanatory power Fact Fallacy Inquiry Opinion Parsimony Premise Propaganda Prudence Reasoning Relevance Rhetoric Rigor Vagueness

Theories of deduction	Constructivism Dialetheism Fictionalism Finitism Formalism Intuitionism Logical atomism Logicism Nominalism Platonic realism Pragmatism Realism

Metalogic and metamathematics

Cantor's theorem Church's theorem Church's thesis Consistency Effective method Foundations of mathematics Gödel's completeness theorem Gödel's incompleteness theorems Soundness Completeness Decidability Interpretation Löwenheim–Skolem theorem Metatheorem Satisfiability Independence Type–token distinction Use–mention distinction

Mathematical logic

General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory

Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus and Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables

Predicate	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Set theory	Set Empty set Enumeration Extensionality Finite set Function Subset Power set Countable set Recursive set Domain Range Ordered pair Uncountable set

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

Non-classical logic

Modal logic	Alethic Axiologic Deontic Doxastic Epistemic Temporal

Intuitionism	Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory

Fuzzy logic	Degree of truth Fuzzy rule Fuzzy set Fuzzy finite element Fuzzy set operations

Substructural logic	Structural rule Relevance logic Linear logic

Paraconsistent logic	Dialetheism

Description logic	Ontology Ontology language

Logicians

Anderson Aristotle Averroes Avicenna Bain Barwise Bernays Boole Boolos Cantor Carnap Church Chrysippus Curry De Morgan Frege Geach Gentzen Gödel Hilbert Kleene Kripke Leibniz Löwenheim Peano Peirce Putnam Quine Russell Schröder Scotus Skolem Smullyan Tarski Turing Whitehead William of Ockham Wittgenstein Zermelo

Lists

Topics	Outline of logic Index of logic articles Mathematical logic Boolean algebra Set theory

Other	Logicians Rules of inference Paradoxes Fallacies Logic symbols

Common logical symbols

& ∨ ¬ ~ → ⊃ ≡ \| ∀ ∃ ⊤ ⊥ ⊢ ⊨ ∴ ∵

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Famous quotes containing the words triple and/or bar:

“The triple pillar of the world transformed
Into a strumpet’s fool.”
—William Shakespeare (1564–1616)

“The bar ... is an exercise in solitude. Above all else, it must be quiet, dark, very comfortable—and, contrary to modern mores, no music of any kind, no matter how faint. In sum, there should be no more than a dozen tables, and a clientele that doesn’t like to talk.”
—Luis Buñuel (1900–1983)