Difference between revisions of "Equivalence relation"
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− | Let <math>S</math> be a set. A 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]]) |
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+ | 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]]) | ||
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Some common examples of equivalence relations: | Some common examples of equivalence relations: | ||
− | * The relation <math>=</math> (equality), on the set of real | + | * The relation <math>=</math> (equality), on the set of [[real number]]s. |
− | * The relation <math>\cong</math> (congruence), on the set of geometric figures in the plane. | + | * 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. | * 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 | + | * 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'']]) |
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)