# Difference between revisions of "Equivalence relation"

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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: | 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: | ||

− | 1. For every element <math>x \in S</math>, <math>x \sim x</math>. (Reflexive property) | + | 1. For every element <math>x \in S</math>, <math>x \sim x</math>. ([[Reflexive property]]) |

− | 2. 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) | + | 2. 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]]) |

− | 3. 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) | + | 3. 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]]) |

## Latest revision as of 11:51, 22 May 2007

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

1. For every element , . (Reflexive property)

2. If such that , then we also have . (Symmetric property)

3. If such that and , then we also have . (Transitive property)

Some common examples of equivalence relations:

- The relation (equality), on the set of real numbers.
- The relation (congruence), on the set of geometric figures in the plane.
- The relation (similarity), on the set of geometric figures in the plane.
- For a given positive integer , the relation , on the set of integers. (Congruence mod
*n*)