Equivalence relation

An equivalence relation is a binary relation on a set that groups elements into categories in which all members are considered "equivalent" under some criterion.

Definition

A relation $\sim$ on set $X$ is a equivalence relation if it satisfies the following properties:

• Reflexivity: $\mathrm{\forall }x\in X$, $x\sim x$.
• Symmetry: $\mathrm{\forall }x,y\in X$ such that $x\sim y$, $y\sim x$.
• Transitivity: $\mathrm{\forall }x,y,z$, if $x\sim y$ and $y\sim z$, then $x\sim z$.

When $x\sim y$, "$x$ is said to be equivalent to $y$" under the relation $\sim$.

Equivalence classes

Given $x\in X$, the equivalence class of $x$, denoted $$[x]=\{y\in X\mid y\sim x\}$$ is the set of elements that are equivalent to $x$.

See also

• Relation
• Quotient set

Terminology

en	fr	de	zh		ja
equivalence relation	relation d'équivalence	Äquivalenzrelation	等价关系	等價關係	同値関係
equivalence class	classe d'équivalence	Äquivalenzklasse	等价类	等價類	同値類
equivalence	équivalence	Äquivalenz	等价	等價	同値
reflexive	réflexif	reflexiv	自反的	自反的	反射的
symmetric	symétrique	symmetrisch	对称的	對稱的	対称的
transitive	transitif	transitiv	传递的	傳遞的	推移的