Definition
A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Equivalently, for all a, b and c in A:
- a ~ a. (Reflexivity)
- if a ~ b then b ~ a. (Symmetry)
- if a ~ b and b ~ c then a ~ c. (Transitivity)
A together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted, is defined as .
Read more about this topic: Equivalence Relation
Famous quotes containing the word definition:
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)