PlanetMath: equivalence relation

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equivalence relation

(Definition)

An equivalence relation $\sim$ on a set is a relation that is:

Reflexive.: $a\sim a$ for all $a\in S$ .
Symmetric.: Whenever $a\sim b$ , then $b\sim a$ .
Transitive.: If $a\sim b$ and $b\sim c$ then $a\sim c$ .

and

are related this way we say that they are equivalent under $\sim$ . If $a\in S$ , then the set of all elements of

that are equivalent to

is called the equivalence class of

An equivalence relation on a set induces a partition on it, and also any partition induces an equivalence relation. Equivalence relations are important, because often the set can be 'transformed' into another set (quotient space) by considering each equivalence class as a single unit.

Two examples of equivalence relations:

1. Consider the set of integers $\mathbbmss{Z}$ and take a positive integer . Then induces an equivalence relation by $a\sim b$ when divides (that is, and leave the same remainder when divided by ).

2. Take a group $(G,\cdot)$ and a subgroup . Define $a\sim b$ whenever $ab^{-1}\in H$ . That defines an equivalence relation. Here equivalence classes are called cosets.