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:
“It is a fundamental characteristic of civilization that man most profoundly mistrusts those living outside his own milieu, so that not only does the Teuton regard the Jew as an incomprehensible and inferior being, but the football player likewise so regards the piano player.”
—Robert Musil (1880–1942)
“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)
“As death, when we come to consider it closely, is the true goal of our existence, I have formed during the last few years such close relations with this best and truest friend of mankind, that his image is not only no longer terrifying to me, but is indeed very soothing and consoling! And I thank my God for graciously granting me the opportunity ... of learning that death is the key which unlocks the door to our true happiness.”
—Wolfgang Amadeus Mozart (1756–1791)