Editing Equivalence relation (section)

== Examples ==

=== Simple example ===

On the set <math>X = \{a, b, c\}</math>, the relation <math>R = \{(a, a), (b, b), (c, c), (b, c), (c, b)\}</math> is an equivalence relation. The following sets are equivalence classes of this relation:
<math display=block>[a] = \{a\}, ~~~~ [b] = [c] = \{b, c\}.</math>

The set of all equivalence classes for <math>R</math> is <math>\{\{a\}, \{b, c\}\}.</math> This set is a [[Partition of a set|partition]] of the set <math>X</math>. It is also called the [[Equivalence relation#Quotient set|quotient set]] of <math>X</math> by <math>R</math>.

=== Equivalence relations ===

The following relations are all equivalence relations:
* "Is equal to" on the set of numbers. For example, <math>\tfrac{1}{2}</math> is equal to <math>\tfrac{4}{8}.</math><ref name=":0" />
* "Is [[Similarity (geometry)|similar]] to" on the set of all [[Triangle (geometry)|triangle]]s.
* "Is [[Congruence (geometry)|congruent]] to" on the set of all [[Triangle (geometry)|triangle]]s.
* Given a [[Function (mathematics)|function]] <math>f:X \to Y</math>, "has the same [[Image (mathematics)|image]] under <math>f</math> as" on the elements of <math>f</math>'s [[domain of a function|domain]] <math>X</math>. For example, <math>0</math> and <math>\pi</math> have the same image under <math>\sin</math>, viz. <math>0</math>. In particular:
** "Has the same absolute value as" on the set of real numbers
** "Has the same cosine as" on the set of all angles.
** Given a natural number <math>n</math>, "is congruent to, [[Modular arithmetic|modulo]] <math>n</math>" on the [[integers]].<ref name=":0" />
** "Have the same length and direction" ([[equipollence (geometry)|equipollence]]) on the set of [[directed line segment]]s.<ref>[[Lena L. Severance]] (1930) [https://babel.hathitrust.org/cgi/pt?id=mdp.39015069379678;view=1up;seq=15 The Theory of Equipollences; Method of Analytical Geometry of Sig. Bellavitis], link from HathiTrust</ref>
** "Has the same birthday as" on the set of all people.

=== Relations that are not equivalences ===

* The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 but not 5 ≥ 7.
* The relation "has a [[common factor]] greater than 1 with" between [[natural numbers]] greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1.
* The [[empty relation]] ''R'' (defined so that ''aRb'' is never true) on a set ''X'' is [[Vacuously true|vacuously]] symmetric and transitive; however, it is not reflexive (unless ''X'' itself is empty).
* The relation "is approximately equal to" between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change. However, if the approximation is defined asymptotically, for example by saying that two functions ''f'' and ''g'' are approximately equal near some point if the limit of ''f − g'' is 0 at that point, then this defines an equivalence relation.