Editing Equivalence relation (section)

=== 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.