World Library  
Flag as Inappropriate
Email this Article

Asymmetric relation

Article Id: WHEBN0001229368
Reproduction Date:

Title: Asymmetric relation  
Author: World Heritage Encyclopedia
Language: English
Subject: Antisymmetric relation, Weak ordering, Binary relation, Asymmetric, Mathematical relations
Collection: Mathematical Relations
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Asymmetric relation

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

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

  • 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

References

  1. ^  .
  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.  Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.