EquivalenceRelation

An equivalence relation ~ on a set X is a RelatioN? satisfying the following conditions: for every a,b,c in X,

    a~a                                 (reflexive property)
    If a~b, then b~a                    (symmetric property)
    If a~b, b~c, then a~c               (transitive property)

Given any x in X, we define the equivalence class of x to be the set [x] = {y in G : x~y}. The set of all such equivalence classes is called the quotient X/~. These form a partition of X, and conversely any partition of X is a quotient X/~ for some equivalence relation ~.

A trivial example of an equivalence relation is equality. X = X/(=) for any set X.