Equivalence relation: Difference between revisions

Latest revision as of 20:53, 10 April 2026

An equivalence relation is a binary relation on a set that groups elements into categories^{[Note 1]} 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: $\forall x \in X$ , $x \sim x$ .
Symmetry: $\forall x, y \in X$ such that $x \sim y$ , $y \sim x$ .
Transitivity: $\forall x, y, z$ , 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 $[x] = {y \in X ∣ y \sim x}$ is the set of elements that are equivalent to $x$ .

Notes

↑ Not to be confused with category in category theory.

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

[1] Not to be confused with category in category theory.

[Note 1]

@@ Line 1: / Line 1: @@
-An '''equivalence relation''' is a [[binary relation]] on a [[set]] that groups elements into categories<ref>Not to be confused with [[category]] in [[category theory]].</ref> in which all members are considered "equivalent" under some criterion.
+An '''equivalence relation''' is a [[binary relation]] on a [[set]] that groups elements into categories<ref group="Note">Not to be confused with [[category]] in [[category theory]].</ref> in which all members are considered "equivalent" under some criterion.
 == Definition ==
@@ Line 8: / Line 8: @@
 * '''Transitivity''': <math>\forall x,y,z</math>, if <math>x\sim y</math> and <math>y\sim z</math>, then <math>x\sim z</math>.
-When <math>x\sim y</math>, then we say that "<math>x</math> is equivalent to <math>y</math>" under the relation <math>\sim</math>.
+When <math>x\sim y</math>, "<math>x</math> is said to be equivalent to <math>y</math>" under the relation <math>\sim</math>.
+== Equivalence classes ==
+Given <math>x \in X</math>, the '''equivalence class''' of <math>x</math>, denoted <math display="block">[x]= \{y \in X \mid y \sim x\}</math> is the set of elements that are equivalent to <math>x</math>.
 == Notes ==
 <references group="Note" />
+== See also ==
+* [[Relation]]
+* [[Quotient set]]
+== Terminology ==
+{{Terminology_table|
+{{Terminology_table/row | equivalence relation | relation d'équivalence | Äquivalenzrelation | 等价关系 | 等價關係 | 同値関係 }}
+{{Terminology_table/row | equivalence class | classe d'équivalence | Äquivalenzklasse | 等价类 | 等價類 | 同値類 }}
+{{Terminology_table/row | equivalence | équivalence | Äquivalenz | 等价 | 等價 | 同値 }}
+{{Terminology_table/row | reflexive | réflexif | reflexiv | 自反的 | 自反的 | 反射的 }}
+{{Terminology_table/row | symmetric | symétrique | symmetrisch | 对称的 | 對稱的 | 対称的 }}
+{{Terminology_table/row | transitive | transitif | transitiv | 传递的 | 傳遞的 | 推移的 }}
+}}