Class Antisymmetric-Relation

Arity: 1
Relation R is an antisymmetric-relation if for distinct x and y, R(x,y) implies not R(y,x). In other words, for all x,y, R(x,y) and R(y,x) => x=y. R(x,x) is still possible.


Instance-Of: Class, Relation, Set
Subclass-Of: Binary-Relation, Relation, Set
Superclass-Of: Asymmetric-Relation, Partial-Order-Relation, Total-Order-Relation


Arity: 2

Implication Axioms for Antisymmetric-Relation:

(=> (Antisymmetric-Relation ?R)
    (=> (And (Holds ?R ?X ?Y) (Holds ?R ?Y ?X)) (= ?X ?Y)))

Equivalence Axioms for Antisymmetric-Relation:

(<=> (Antisymmetric-Relation ?R)
     (And (Binary-Relation ?R)
          (=> (And (Holds ?R ?X ?Y) (Holds ?R ?Y ?X)) (= ?X ?Y))))

Implication Axioms mentioning Antisymmetric-Relation:

(=> (Asymmetric-Relation ?R) (Antisymmetric-Relation ?R))

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

Equivalence Axioms mentioning Antisymmetric-Relation:

(<=> (Asymmetric-Relation ?R)
     (And (Antisymmetric-Relation ?R) (Irreflexive-Relation ?R)))

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