Editing Equivalence relation (section)

== Comparing equivalence relations ==
{{See also|Partition of a set#Refinement of partitions}}
If <math>\sim</math> and <math>\approx</math> are two equivalence relations on the same set <math>S</math>, and <math>a \sim b</math> implies <math>a \approx b</math> for all <math>a, b \in S,</math> then <math>\approx</math> is said to be a '''coarser''' relation than <math>\sim</math>, and <math>\sim</math> is a '''finer''' relation than <math>\approx</math>.  Equivalently,
* <math>\sim</math> is finer than <math>\approx</math> if every equivalence class of <math>\sim</math> is a subset of an equivalence class of <math>\approx</math>, and thus every equivalence class of <math>\approx</math> is a union of equivalence classes of <math>\sim</math>.
* <math>\sim</math> is finer than <math>\approx</math> if the partition created by <math>\sim</math> is a refinement of the partition created by <math>\approx</math>.

The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest.

The relation "<math>\sim</math> is finer than <math>\approx</math>" on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a [[geometric lattice]].<ref>{{citation|title=Lattice Theory|volume=25|series=Colloquium Publications|publisher=American Mathematical Society|first=Garrett|last=Birkhoff|author-link=Garrett Birkhoff|edition=3rd|year=1995|isbn=9780821810255}}. Sect. IV.9, Theorem 12, page 95</ref>