Equivalence Relation - Definition

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:

    Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.
    The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    One definition of man is “an intelligence served by organs.”
    Ralph Waldo Emerson (1803–1882)

    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)