Difference between revisions of "Equivalence relation"

Latest revision as of 11:17, 15 February 2025

Let $S$ be a set. A binary relation $\sim$ on $S$ is said to be an equivalence relation if $\sim$ satisfies the following three properties:

For every element $x \in S$ , $x \sim x$ . (Reflexive property)
If $x, y \in S$ such that $x \sim y$ , then we also have $y \sim x$ . (Symmetric property)
If $x, y, z \in S$ such that $x \sim y$ and $y \sim z$ , then we also have $x \sim z$ . (Transitive property)

Some common examples of equivalence relations:

The relation $=$ equality, on the set of complex numbers.
The relation $\cong$ (congruence), on the set of geometric figures in the plane.
The relation $\sim$ (similarity), on the set of geometric figures in the plane.
For a given positive integer $n$ , the relation $\equiv \pmod n$ , on the set of integers. (Congruence modulo n)

This article is a stub. Help us out by expanding it.

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 Let <math>S</math> be a [[set]].  A [[binary relation]] <math>\sim</math> on <math>S</math> is said to be an '''equivalence relation''' if <math>\sim</math> satisfies the following three properties:
+# For every element <math>x \in S</math>, <math>x \sim x</math>.  ([[Reflexive property]])
-. For every element <math>x \in S</math>, <math>x \sim x</math>.  ([[Reflexive property]])
+# If <math>x, y \in S</math> such that <math>x \sim y</math>, then we also have <math>y \sim x</math>.  ([[Symmetric property]])
+# If <math>x, y, z \in S</math> such that <math>x \sim y</math> and <math>y \sim z</math>, then we also have <math>x \sim z</math>.  ([[Transitive property]])
-. If <math>x, y \in S</math> such that <math>x \sim y</math>, then we also have <math>y \sim x</math>.  ([[Symmetric property]])
-. If <math>x, y, z \in S</math> such that <math>x \sim y</math> and <math>y \sim z</math>, then we also have <math>x \sim z</math>.  ([[Transitive property]])
 Some common examples of equivalence relations:
-* The relation <math>=</math> (equality), on the set of [[real number]]s.
+* The relation <math>=</math> [[equality]], on the set of [[complex number]]s.
 * The relation <math>\cong</math> (congruence), on the set of geometric figures in the [[plane]].
 * The relation <math>\sim</math> (similarity), on the set of geometric figures in the plane.
-* For a given [[positive integer]] <math>n</math>, the relation <math>\equiv \pmod n</math>, on the set of [[integer]]s.  ([[Congruence]] [[Modular arithmetic|mod ''n'']])
+* For a given [[positive integer]] <math>n</math>, the relation <math>\equiv \pmod n</math>, on the set of [[integer]]s.  ([[Congruence]] [[Modular arithmetic|modulo ''n'']])
+{{stub}}

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Difference between revisions of "Equivalence relation"

Latest revision as of 11:17, 15 February 2025