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 .

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