Partition of A Set - Partitions and Equivalence Relations

Partitions and Equivalence Relations

For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from any partition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the same part in P. Thus the notions of equivalence relation and partition are essentially equivalent.

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Famous quotes containing the words partitions and/or relations:

    “Walls have cracks and partitions ears.”
    —Chinese proverb.

    “Consciousness, we shall find, is reducible to relations between objects, and objects we shall find to be reducible to relations between different states of consciousness; and neither point of view is more nearly ultimate than the other.”
    —T.S. (Thomas Stearns)