Editing Asymmetric relation (section)

=== Definition ===

The binary relation <math>R</math> is called '''{{em|asymmetric}}''' if for all <math>a, b \in X,</math> if <math>a R b</math> is true then <math>b R a</math> is false; that is, if <math>(a, b) \in R</math> then <math>(b, a) \not\in R.</math> 
This can be written in the notation of [[first-order logic]] as
<math display=block>\forall a, b \in X: a R b \implies \lnot(b R a).</math>
A [[Logical equivalence|logically equivalent]] definition is:
:for all <math>a, b \in X,</math> at least one of <math>a R b</math> and <math>b R a</math> is {{em|false}},
which in first-order logic can be written as:
<math display=block>\forall a, b \in X: \lnot(a R b \wedge b R a).</math>
A relation is asymmetric if and only if it is both [[Antisymmetric relation|antisymmetric]] and [[Reflexive relation|irreflexive]],<ref>{{citation|first1=Yves|last1=Nievergelt|title=Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography|publisher=Springer-Verlag|year=2002|page=[https://books.google.com/books?id=_H_nJdagqL8C&pg=PA158 158]}}.</ref> so this may also be taken as a definition.