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:

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

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.

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