Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

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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 equivalence classes form a partition of X, and conversely every partition of x corresponds to some unique equivalence relation ~. The set of all equivalence classes is denoted X/~, and called a quotient set. In cases where X has some additional structure preserved under ~, the quotient naturally becomes an object of the same type; the map that sends x to [x] is then a HomoMorphism.