Equivalence Relation - Fundamental Theorem of Equivalence Relations

Fundamental Theorem of Equivalence Relations

A key result links equivalence relations and partitions:

  • An equivalence relation ~ on a set X partitions X.
  • Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection from the set of all possible equivalence relations on X and the set of all partitions of X.

Read more about this topic:  Equivalence Relation

Famous quotes containing the words fundamental, theorem and/or relations:

    One of the fundamental reasons why so many doctors become cynical and disillusioned is precisely because, when the abstract idealism has worn thin, they are uncertain about the value of the actual lives of the patients they are treating. This is not because they are callous or personally inhuman: it is because they live in and accept a society which is incapable of knowing what a human life is worth.
    John Berger (b. 1926)

    To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.
    Albert Camus (1913–1960)

    The interest in life does not lie in what people do, nor even in their relations to each other, but largely in the power to communicate with a third party, antagonistic, enigmatic, yet perhaps persuadable, which one may call life in general.
    Virginia Woolf (1882–1941)