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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, |
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An equivalence relation ~ on a SeT X is a MathematicalRelation satisfying the following conditions: for every a,b,c in X, |
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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 ~. |
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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 ~. |
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A trivial example of an equivalence relation is equality. X = X/(=) for any set X. |
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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. Equality always defines an equivalence relation, and in fact it is minimal. X = X/=. |
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