Equivalence relation

In mathematics, an equivalence relation formula_1 on a set is a mathematical relation that is symmetric, transitive and reflexive. For a given element formula_2 on that set, the set of all elements related to formula_2 (in the sense of formula_1) is called the equivalence class of formula_2, and written as formula_6. 
With an equivalence relation, it is possible to partition a set into distinct equivalence classes. As an example, consider the set of all animals on a farm and define the following relation: two animals are related if they belong to the same species. Under this relation, a cow is related to an ox, but not to a chicken. In fact, this relation is an equivalence relation because:
In this example, the set of all animals related to a particular ox forms an equivalence class—it is the set of cattle. In fact, the set of all animals on this farm can be partitioned into different equivalence classes (in this case species).