Equivalence relation: Difference between revisions
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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 == | == Definition == | ||
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 
on set 
is a equivalence relation if it satisfies the following properties:
- Reflexivity:
, 
. - Symmetry:
such that 
, 
. - Transitivity:
, if and 
, then 
.
When , "
is said to be equivalent to 
" under the relation .
Equivalence classes
Given 
, the equivalence class of , denoted
is the set of elements that are equivalent to .
Notes
- ↑ Not to be confused with category in category theory.
See also
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 |