Asymmetric relation

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In mathematics an asymmetric relation is a binary relation on a set X where:

  • For all a and b in X, if a is related to b, then b is not related to a.[1]

In mathematical notation, this is:

\forall a, b  \in X,\ a R b \; \Rightarrow \lnot(b R a).


Examples[edit]

An example is < (less-than): if x < y, then necessarily y is not less than x. In fact, one of Tarski's axioms characterizing the real numbers R is that < over R is asymmetric.

An asymmetric relation need not be total. For example, strict subset or ⊊ is asymmetric, and neither of the sets {1,2} and {3,4} is a strict subset of the other. In general, every strict partial order is asymmetric, and conversely, every transitive asymmetric relation is a strict partial order.

Not all asymmetric relations are strict partial orders, however. An example of an asymmetric intransitive relation is the rock-paper-scissors relation: if X beats Y, then Y does not beat X, but no one choice wins all the time.

The ≤ (less than or equal) operator, on the other hand, is not asymmetric, because reversing x ≤ x produces x ≤ x and both are true. In general, any relation in which x R x holds for some x (that is, which is not irreflexive) is also not asymmetric.

Asymmetric is not the same thing as "not symmetric": a relation can be neither symmetric nor asymmetric, such as ≤, or can be both, only in the case of the empty relation (vacuously).

Properties[edit]

  • A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[2]
  • Restrictions and inverses 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 a R b and b R a, transitivity gives a R a, contradicting irreflexivity.

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. Prague: School of Mathematics - Physics Charles University. p. 1.  Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".