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.
Read more about this topic: Partition Of A Set
Famous quotes containing the words partitions and/or relations:
“Great wits are sure to madness near allied,
And thin partitions do their bounds divide.”
—John Dryden (1631–1700)
“Actually, the laboring man has not leisure for a true integrity day by day; he cannot afford to sustain the manliest relations to men; his labor would be depreciated in the market.
He has no time to be anything but a machine.”
—Henry David Thoreau (1817–1862)