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Relation Properties

Definition: A binary relation R on a set S is reflexive, symmetric, respectively transitive, if it satisfies the following properties:

R is reflexive on S : <=>

forall x in S: <x, x> in R;

R is symmetric on S : <=>

forall x, y: <x, y> in R => <y, x> in R;

R is transitive on S : <=>

forall x, y, z: (<x, y> in R /\  <y, z> in R) => <x, z> in R.

Example: equality is reflexive, symmetric and transitive on every set.