Properties
Every element x of X is a member of the equivalence class . Every two equivalence classes and are either equal or disjoint. Therefore, the set of all equivalence classes of X forms a partition of X: every element of X belongs to one and only one equivalence class. Conversely every partition of X comes from an equivalence relation in this way, according to which x ~ y if and only if x and y belong to the same set of the partition.
It follows from the properties of an equivalence relation that
-
- x ~ y if and only if = .
In other words, if ~ is an equivalence relation on a set X, and x and y are two elements of X, then these statements are equivalent:
- .
Read more about this topic: Equivalence Class
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)