Equivalence relation - Revision history

InfernalAtom683 at 12:53, 10 April 2026

2026-04-10T12:53:57Z

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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 ==

InfernalAtom683 at 12:52, 10 April 2026

2026-04-10T12:52:48Z

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	== Notes ==		== Notes ==
	<references group="Note" />		<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 }}
			}}

InfernalAtom683 at 05:34, 25 March 2026

2026-03-25T05:34:32Z

← Older revision		Revision as of 13:34, 25 March 2026
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	* '''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>.		* '''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 ==		== Notes ==
	<references group="Note" />		<references group="Note" />

InfernalAtom683: Created page with "An '''equivalence relation''' is a binary relation on a set that groups elements into categoriesNot to be confused with category in category theory. 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'..."

2026-03-22T13:19:58Z

Created page with "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. == Definition == A relation <math>\sim</math> on set <math>X</math> is a equivalence relation if it satisfies the following properties: * '''Reflexivity''': <math>\forall x\in X</math>, <math>x\sim x</math>. * '''Symmetry'..."

New page

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.

== Definition ==
A [[relation]] <math>\sim</math> on set <math>X</math> is a equivalence relation if it satisfies the following properties:

* '''Reflexivity''': <math>\forall x\in X</math>, <math>x\sim x</math>.
* '''Symmetry''': <math>\forall x,y\in X</math> such that <math>x\sim y</math>, <math>y\sim x</math>.
* '''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>.

== Notes ==
<references group="Note" />