Asymmetric relation

In mathematics, an asymmetric relation is a binary relation on a set where for all if is related to then is not related to [1]

Formal definitionEdit

A binary relation on   is any subset   of   Given   write   if and only if   which means that   is shorthand for   The expression   is read as "  is related to   by  " The binary relation   is called asymmetric if for all   if   is true then   is false; that is, if   then   This can be written in the notation of first-order logic as

 

A logically equivalent definition is:

for all   at least one of   and   is false,

which in first-order logic can be written as:

 

An example of an asymmetric relation is the "less than" relation   between real numbers: if   then necessarily   is not less than   The "less than or equal" relation   on the other hand, is not asymmetric, because reversing for example,   produces   and both are true. Asymmetry is not the same thing as "not symmetric": the less-than-or-equal relation is an example of a relation that is neither symmetric nor asymmetric. The empty relation is the only relation that is (vacuously) both symmetric and asymmetric.

PropertiesEdit

  • A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[2]
  • Restrictions and converses of asymmetric relations are also asymmetric. For example, the restriction of   from the reals to the integers is still asymmetric, and the inverse   of   is also asymmetric.
  • A transitive relation is asymmetric if and only if it is irreflexive:[3] if   and   transitivity gives   contradicting irreflexivity.
  • As a consequence, a relation is transitive and asymmetric if and only if it is a strict partial order.
  • Not all asymmetric relations are strict partial orders. An example of an asymmetric non-transitive, even antitransitive relation is the rock paper scissors relation: if   beats   then   does not beat   and if   beats   and   beats   then   does not beat  
  • An asymmetric relation need not have the connex property. For example, the strict subset relation   is asymmetric, and neither of the sets   and  is a strict subset of the other. A relation is connex if and only if its complement is asymmetric.

See alsoEdit

ReferencesEdit

  1. ^ Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer-Verlag, p. 273.
  2. ^ Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer-Verlag, p. 158.
  3. ^ Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School of Mathematics - Physics Charles University. p. 1. Archived from the original (PDF) on 2013-11-02. Retrieved 2013-08-20. Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".