Talk:Asymmetric relation

nonsymmetric

Someone redirected nonsymmetric relation to asymmetric. However, they deleted the article without including anything about nonsymmetric relations. So here is the deleted part. Apparently he thought it look wierd and deleted it. Way to go. That's a great way to do things. Gregbard 23:04, 29 August 2007 (UTC)

Deleted Part

In logic, the nonsymmetrical or partimsymmetric relation is defined as:

(x)(y)(Rxy$\to$(($\exists$x)($\exists$y)(Ryx)$\wedge$($\exists$x)($\exists$y)(~Ryx))

Examples

"x loves y", "is the brother of"

See also

• symmetric relation
• asymmetric relation
• antisymmetric relation

True there is currently nothing in asymmetric relation that explains what a nonsymmetric relation is. I think this needs to be remedied. Asymmetric relation has two definitions; which one of these, if any, corresponds with the definition of a nonsymmetric relation? I too dislike redirects to articles that have no mention at all of the first article.

Gregbard, the notation used in your formula looks rather non-standard (the universal quantifier seems to be implicit), and even so I don't think the formula makes sense because you are quantifying over the same variables twice.

My searching reveals that the matter was discussed in Wikipedia talk:WikiProject Mathematics/Archive 28#Negation of definitions

--Egriffin (talk) 17:13, 7 September 2008 (UTC)

nonsymmetric

nonsymmetric relation and asymmetric relation is the same.

In Logic, the non-symmetric or the asymmetric relation occurs when, in the same context U (being U a finite set; and x and y elements of U), some couples <x,y> of R do exist and <y,x>, too; there are also cases where <x,y> do exist but <y,x> does not. From the point of view of symmetry, any pattern is possible for a given couple. The non-symmetric property of xRy is defined taken advantage of NOR connector (either symmetric or antisymmetric but neither both nor none) as:

(x)(y)(xRy$\to$(($\exists$x)($\exists$y)(yRx)NOR($\exists$x)($\exists$y)(~yRx))

Examples

"x loves y", "x is the brother of y"

See also

• symmetric relation
• antisymmetric relation
• homegeneous relation

Anti-transitivity

Ajackson716 justified their recent edit by "See Intransitivity#Antitransitivity for details regarding the impossibility of a simultaneously transitive and anti-transitive relation. Additionally, it is not even possible for a relation to be both transitive and non-transitive". Here is my justification for reverting it:

I agree with the 2nd sentence. As for the 1st, antitransitivity ($\forall a,b,c:aRb\land bRc\implies \lnot aRc$) is different from non-transitivity ($\lnot \forall a,b,c:aRb\land bRc\implies aRc$). A relation is both anti-transitive and transitive iff $\forall a,b,c:\lnot (aRb\land bRc)$; an example is $\{(0,1),(2,3)\}$ in $\{0,1,2,3\}$; such a relation of course needs to be asymmetric. - Jochen Burghardt (talk) 12:32, 26 October 2025 (UTC)

Hi @Jochen Burghardt - thanks for the example! It is however only vacuously true, as it contains two mutually exclusive mappings. And while technically correct may be the best kind of correct, I think it is misleading to describe a relational property (asymmetry) in terms of vacuous conditions. Someone trying to understand asymmetry is best served by grasping the necessary and sufficient conditions of irreflexivity and anti-symmetry, along with relevant implications, e.g. irreflexivity and transitivity.

Including joint transitivity and anti-transitivity in the same list, without mentioning their vacuousness, is tantamount to including the empty set in almost every list of sufficient conditions. While true, it does little to illuminate the underlying principle being described. Obviously your example is valid, so I can no longer justify removing it entirely.

Instead may I suggest that you incorporate a mention of its vacuousness, and provide an example such as the one you have included here? At least from my perspective, seeing it listed alongside the other conditions, without qualification, is confusing. Ajackson716 (talk) 07:45, 27 October 2025 (UTC)

The relations satisfying $\forall a,b,c:\lnot (aRb\land bRc)$ are an interesting class by its own, larger than e.g. the set of coreflexive relations (on a finite universe set). The fact that they vacuously satisfy transitivity (and antitransitivity) doesn't make the class itself trivial. I agree that their sufficiency for asymmetry should not be reported in the introduction or definition. However, it is a valid property and not more trivial than many others listed in the bottom section "Properties" (all proofs there are pretty elementary and easy to find), so it should be kept there, imo. - Jochen Burghardt (talk) 16:49, 27 October 2025 (UTC)

Thanks for the response. I see what you're saying, but I still think it's confusing. The reason I edited in the first place was the result of significant time spent trying to find information about these cases, or examples, and coming up empty. If the class of such relations is interesting enough within itself, I think it would be worth mentioning that along with the condition, and perhaps linking to an article on those specific classes. This article is about Asymmetry, not classes of vacuously transitive and anti-transitive relations, which, while perhaps not trivial, are certainly not intuitively realizable as a key component of asymmetry. To me it seems that asymmetry is simply a property that necessarily falls out of such classes, rather than such classes illuminating the property of asymmetry itself. Either way, I'll stop poking at it now. Thanks for taking the time to explain. Ajackson716 (talk) 19:26, 27 October 2025 (UTC)