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:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)