Asymmetric relation

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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 definition[edit]

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.

Properties[edit]

  • 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 also[edit]

References[edit]

  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".