blanketglossary

Equivalence relation

Definition

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. A simpler example is numerical equality. Any number is equal to itself (reflexive). If , then (symmetric). If and , then (transitive).

Related concepts

Abraham Fraenkel Abstract logic Ackermann set theory Aleph number Alexander Bogomolny Algebraic expression Algebraic logic Algebraic structure Almost Alphabet (formal languages)Alternative set theory Amorphous set Antisymmetric relation Apartness relation Argument Arity Associativity Asymmetric relation Atomic formula Atomic model (mathematical logic)Atomic sentence Automata theory Automated theorem proving Automorphism group Axiom Axiom of adjunction Axiom of choice Axiom of constructibility Axiom of countable choice Axiom of dependent choice Axiom of determinacy Axiom of extensionality Axiom of global choice Axiom of infinity Axiom of limitation of size Axiom of pairing Axiom of power set Axiom of projective determinacy Axiom of regularity Axiom of union Axiom schema Axiom schema of replacement Axiom schema of specification Axiomatic system Axiomatization of Boolean algebras Banach–Tarski paradox Bas van Fraassen Bell number Bertrand Russell Bijection Bijections Bijective Bijective function Binary operation Binary relation Boolean algebra Boolean algebras canonically defined Boolean function Borel equivalence relation Burali-Forti paradox Cantor's diagonal argument Cantor's paradox Cantor's theorem Cardinal number Cardinality Cartesian product Cartesian square Categorical theory Category (mathematics)Category of sets Category theory Church encoding Church–Turing thesis Class (set theory)Classical logic Cluster graph Codomain Common factor Compactness theorem Complement (set theory)Complete lattice Complete theory Composition of relations Computability theory Computable function Computable set Computably enumerable set Concrete category Congruence (geometry)Congruence relation Conjugacy class Connected relation Conservative extension Consistency Constructible universe Construction of the real numbers Constructive mathematics Constructive set theory Continuum hypothesis Coset Countable set Cut-the-knot De Morgan's laws Decidability (logic)Decision problem Dedekind-infinite set Deductive system Dependency relation Diagram (mathematical logic)Directed graph Directed line segment Disjoint union Dobinski's formula Doctrine of equivalents Domain of a function Element (mathematics)Elementary diagram Elementary equivalence Elementary function arithmetic Empty relation Empty set Encyclopedia of Mathematics Enumeration Equality (mathematics)Equiconsistency Equipollence (geometry)Equivalence (disambiguation)Equivalence class Ernst Zermelo Euclid's Elements Euclidean geometry Euclidean relation European Mathematical Society Existential quantification Expression (mathematics)Extension by definitions Extension by new constant and function names Extensionality Family of sets Filter on a set Finitary relation Finite-valued logic Finite model theory Finite set First-order logic Fixed-point logic Fixed point Forcing (mathematics)Formal grammar Formal language Formal proof Formal semantics (logic)Formal system Formation rule Foundations of geometry Foundations of mathematics Free logic Free object Free variables and bound variables Function (mathematics)Function composition Functional predicate Fuzzy set Garrett Birkhoff General set theory Geometric lattice Georg Cantor Grothendieck universe Ground expression Ground formula Group (mathematics)Group action Group theory Groupoid Gödel's completeness theorem Gödel's incompleteness theorems Gödel numbering Halting problem Hereditarily finite set Hereditary set Higher-order logic Hilbert's axioms Hilbert system History of logic History of mathematical logic Homeomorphism Homogeneous relation Hyperfinite equivalence relation Identity function Identity relation If and only if Image (mathematics)Inaccessible cardinal Independence (mathematical logic)Inference Infimum Infinite-valued logic Infinite set Information theory Inhabited set Injective Injective function Integers Interpretation (logic)Interpretation (model theory)Interpretation function Intersection (set theory)Invariant (mathematics)Inverse function Isomorphism John Lucas (philosopher)John Wiley & Sons John von Neumann Join-semilattice Join (mathematics)Join and meet Karel Hrbacek Katherine Brading Kernel (algebra)Kolmogorov complexity Kripke's theory of truth Kripke–Platek set theory Kurt Gödel Lambda calculus Large cardinal Lattice (order)Law of excluded middle Lemma (mathematics)Lena L. Severance Lindström's theorem List of Hilbert systems List of axioms List of first-order theories List of formal systems List of mathematical theories List of set identities and relations List of statements independent of ZFC Logic Logical biconditional Logical conjunction Logical connective Logical consequence Logical constant Logical disjunction Logical equality Logical equivalence Logical matrix Logical truth Logicism Löwenheim–Skolem theorem Many-valued logic Map (mathematics)Marcel Dekker Martin's axiom Material conditional Mathematical logic Mathematical object Mathematics Meet-semilattice Meet (mathematics)Metalanguage Minimal axioms for Boolean algebra Model complete theory Model theory Modular arithmetic Monadic predicate calculus Monadic second-order logic Monoid Morley's categoricity theorem Morse–Kelley set theory NP (complexity)Naive set theory Natural deduction Natural number Natural numbers Negation New Foundations Non-Euclidean geometry Non-logical symbol Non-standard model Non-standard model of arithmetic Normal subgroup Open formula Operation (mathematics)Orbit (group theory)Order relation Ordered pair Ordinal analysis Ordinal number P (complexity)P versus NP problem Pairwise disjoint Paradoxes of set theory Partial equivalence relation Partial order Partially ordered set Partition of a set Partitions of a set Paul Bernays Paul Cohen Peano axioms Permutation Permutation group Philosophy of mathematics Power set Predicate (mathematical logic)Predicate logic Predicate variable Preimage Preorder Prewellordering Prime model Primitive recursive arithmetic Primitive recursive function Principia Mathematica Projection (relational algebra)Projection (set theory)Proof of impossibility Proof theory Proposition Propositional calculus Propositional formula Propositional variable Quantifier (logic)Quantifier rank Quotient by an equivalence relation Quotient set Quotient space (topology)Raymond Wilder Recursion Recursive set Reflexive relation Relation (mathematics)Relational algebra Reverse mathematics Richard Dedekind Robert Dilworth Robinson arithmetic Rule of inference Russell's paradox Satisfiability Saturated model Saunders Mac Lane Schröder–Bernstein theorem Second-order arithmetic Second-order logic Self-verifying theories Semantic theory of truth Semantics of logic Sentence (mathematical logic)Sequent calculus Set-builder notation Set (mathematics)Set inclusion Set theory Setoid Signature (logic)Similarity (geometry)Singleton (mathematics)Skolem arithmetic Soundness Spectrum of a sentence Spectrum of a theory Square of opposition Strength (mathematical logic)Strict partial order Strict total order String (formal languages)Structure (mathematical logic)Subgroup Subset Substitution (algebra)Substitution (logic)Substructure (mathematics)Supertask Supremum Surjection Surjective Surjective function Suslin's problem Syllogism Symbol (formal)Symmetric difference Symmetric relation Syntax (logic)T-schema Tarski's axiomatization of the reals Tarski's axioms Tarski's theory of truth Tarski's undefinability theorem Tarski–Grothendieck set theory Tautology (logic)Term (logic)Term logic Ternary equivalence relation Theorem Theories of truth Theory (mathematical logic)Thomas Jech Thoralf Skolem Three-valued logic Timeline of mathematical logic Tolerance relation Topological conjugacy Topological space Torus Total order Total preorder Total relation Transfer principle Transfinite induction Transformation group Transitive relation Transitive set Triangle (geometry)True arithmetic Truth predicate Truth table Truth value Tuple Turing machine Type (model theory)Type theory Ultrafilter (set theory)Ultrafilter on a set Ultraproduct Uncountable set Undecidable problem Uninterpreted function Union (set theory)Uniqueness quantification Universal quantification Universal set Universe (mathematics)Up to Urelement Vacuously true Validity (logic)Variable (mathematics)Venn diagram Von Neumann universe Von Neumann–Bernays–Gödel set theory Weak ordering Well-defined Well-formed formula Well-founded relation Well-order Well-quasi-ordering Willard Van Orman Quine Zermelo set theory Zermelo–Fraenkel set theory

148 concepts already in your glossary