If and are related this way we say that they are equivalent under . If , then the set of all elements of that are equivalent to is called the equivalence class of . The set of all equivalence classes under is written .
An equivalence relation on a set induces a partition on it. Conversely, 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 and take a positive integer . Then induces an equivalence relation by when divides (that is, and leave the same remainder when divided by ).
2. Take a group and a subgroup . Define whenever . That defines an equivalence relation. Here equivalence classes are called cosets.
|Date of creation||2013-03-22 11:48:27|
|Last modified on||2013-03-22 11:48:27|
|Last modified by||CWoo (3771)|