ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D178
Equivalence relation

A D4424: Binary endorelation $(X \times X, R)$ is an equivalence relation if and only if
 (1) $$\forall \, x \in X : (x, x) \in R$$ D287: Reflexive binary relation (2) $$\forall \, x, y \in X \left( (x, y) \in R \quad \implies \quad (y, x) \in R \right)$$ D294: Symmetric binary relation (3) $$\forall \, x, y, z \in R \left( (x, y), (y, z) \in R \quad \implies \quad (x, z) \in R \right)$$ D288: Transitive binary relation

A D4424: Binary endorelation $(X \times X, R)$ is an equivalence relation if and only if
 (1) $$\forall \, x \in X : x R x$$ D287: Reflexive binary relation (2) $$\forall \, x, y \in X \left( x R y \quad \implies \quad y R x \right)$$ D294: Symmetric binary relation (3) $$\forall \, x, y, z \in R \left( x R y, y R z \quad \implies \quad x R z \right)$$ D288: Transitive binary relation