# Equivalence relation facts for kids

Kids Encyclopedia Facts

In mathematics, an equivalence relation $R$ on a set is a mathematical relation that is symmetric, transitive and reflexive. For a given element $a$ on that set, the set of all elements related to $a$ (in the sense of $R$) is called the equivalence class of $a$, and written as $[a]$.

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:

• It is reflexive: each animal is of the same species as itself
• It is symmetric: if a first animal is in the same species as a second animal, then the second animal is also in the same species as the first animal.
• It is transitive: if a first animal is in the same species as a second animal, and the second animal is in the same species as a third animal, then the first animal is in the same species as the third animal.

In this example, the set of all animals related to a particular ox forms an equivalence classâ€”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).

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Equivalence relation Facts for Kids. Kiddle Encyclopedia.