Equivalence relation

A relation is between two given sets. So a relation R between set A and a set B is a subset of their cartesian product:

$R\subseteq A\times B$

An equivalence relation in a set A is a relation $R\subseteq A\times A$ i.e. an endo-relation in a set, which obeys the conditions:

• reflexivity
• symmetry
• transitivity

An example of this is a sum fractional numbers. Here a rational number can be represented as several different fractions with different denominators, so by making the fractions have a common denominator we can simplify the addition.