Class Reflexive-Relation

Arity: 1
Documentation: Relation R is reflexive if R(x,x) for all x in the domain of R.
Instance-Of: Class, Relation, Set
Subclass-Of: Binary-Relation, Relation, Set
Superclass-Of: Equivalence-Relation, Partial-Order-Relation, Total-Order-Relation


Arity: 2

Implication Axioms for Reflexive-Relation:

(=> (Reflexive-Relation ?R)
    (=> (Instance-Of ?X (Exact-Domain ?R)) (Holds ?R ?X ?X)))

Equivalence Axioms for Reflexive-Relation:

(<=> (Reflexive-Relation ?R)
     (And (Binary-Relation ?R)
          (=> (Instance-Of ?X (Exact-Domain ?R)) (Holds ?R ?X ?X))))

Implication Axioms mentioning Reflexive-Relation:

(=> (Equivalence-Relation ?R) (Reflexive-Relation ?R))

(=> (Partial-Order-Relation ?R) (Reflexive-Relation ?R))

Equivalence Axioms mentioning Reflexive-Relation:

(<=> (Equivalence-Relation ?R)
     (And (Reflexive-Relation ?R)
          (Symmetric-Relation ?R)
          (Transitive-Relation ?R)))

(<=> (Partial-Order-Relation ?R)
     (And (Reflexive-Relation ?R)
          (Antisymmetric-Relation ?R)
          (Transitive-Relation ?R)))