Class Binary-Relation

Subclass-Of@Frame-Ontology: Relation, Set
Superclass-Of@Frame-Ontology:
Many-To-Many-Relation@Frame-Ontology, Many-To-One-Relation@Frame-Ontology, One-To-Many-Relation@Frame-Ontology, Transitive-Relation@Frame-Ontology, Weak-Transitive-Relation@Frame-Ontology ...
Has-Instance@Frame-Ontology:
Disjoint, Generalized-Intersection, Generalized-Union, Proper-Subset, Subset ...
Instance-Of@Frame-Ontology: Class@Frame-Ontology, Relation, Set
Domain-Of@Frame-Ontology: Inverse, Composition-Of@Frame-Ontology, Domain@Frame-Ontology
Range-Of@Frame-Ontology:
Inverse, Compose@Frame-Ontology, Has-One@Ol-User%Slot-Constraint-Sugar, Visible-Slots-For-Query@Interface-Ontology
Arity@Frame-Ontology: 1
Inherited-Slot-Value@Frame-Ontology: 2
Documentation@Ol%Frame-Ontology:
A binary relation maps instances of a class to instances of another class. Its arity is 2. Binary relations are often shown as slots in frame systems.

Notes:


Slots:

Domain@Frame-Ontology:
Composition-Of@Frame-Ontology:
Inverse:
Arity@Frame-Ontology: 2

Implication Axioms for Binary-Relation:

(=> (Binary-Relation ?Relation)
    (Forall (?Tuple) (=> (Member ?Tuple ?Relation) (Double ?Tuple))))

(=> (Binary-Relation ?Relation) (Not (Empty ?Relation)))

(=> (Binary-Relation ?Relation)
    (= (Arity@Frame-Ontology ?Relation) 2))


Equivalence Axioms for Binary-Relation:

(<=> (Binary-Relation ?Relation)
     (And (Relation ?Relation)
          (Not (Empty ?Relation))
          (Forall (?Tuple)
                  (=> (Member ?Tuple ?Relation) (Double ?Tuple)))))


Implication Axioms mentioning Binary-Relation:

(=> (One-One ?R) (Binary-Relation ?R))

(=> (Many-One ?R) (Binary-Relation ?R))

(=> (One-Many ?R) (Binary-Relation ?R))

(=> (Many-Many ?R) (Binary-Relation ?R))

(=> (Unary-Function ?F) (Binary-Relation ?F))


Equivalence Axioms mentioning Binary-Relation:

(<=> (One-One ?R)
     (And (Binary-Relation ?R)
          (Function ?R)
          (Value-Type@Frame-Ontology ?R Inverse Function)
          (Value-Cardinality@Frame-Ontology ?R Inverse 1)))

(<=> (Many-One ?R) (And (Binary-Relation ?R) (Function ?R)))

(<=> (One-Many ?R)
     (And (Binary-Relation ?R)
          (Value-Type@Frame-Ontology ?R Inverse Function)
          (Value-Cardinality@Frame-Ontology ?R Inverse 1)))

(<=> (Many-Many ?R)
     (And (Binary-Relation ?R)
          (Not (Function ?R))
          (Not (Function (Inverse ?R)))))

(<=> (Unary-Function ?F) (And (Function ?F) (Binary-Relation ?F)))


Axioms mentioning Binary-Relation:

(Nth-Domain@Frame-Ontology Composition 3 Binary-Relation)

(Nth-Domain@Frame-Ontology Composition 2 Binary-Relation)

(Nth-Domain@Frame-Ontology Composition 1 Binary-Relation)

(Nth-Domain@Frame-Ontology Value-Cardinality@Frame-Ontology
            2
            Binary-Relation)

(Nth-Domain@Frame-Ontology Slot-Cardinality@Frame-Ontology
            2
            Binary-Relation)