Talk:Antisymmetric relation

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[edit] Regarding the terms "equality" and "antisymmetry"

I believe there is a cycle in the definitions: Equality is defined as binary relation which is reflexive, symmetric transitive and antisymmetric. The definition of antisymmetry refers to the notion of equality (a R b and b R a => a = b). I don't know how to fix this.

cheers, chris

The '=' above is identity, not equality. Identity is primitive in metaphysics and logic, but is characterised by Leibniz's Law. The reference to equality in the article is more strictly speaking equivalence; two objects can be equivalent without being identical (consider the universal equivalence relation: \forall a, b \in X, a R b). Acanon 20:43, 20 Sep 2004 (UTC)

prakash kumar devta from IIITB

I would answer this by saying that the axiom of antisymmetry is an axiom of first-order logic with equality. So equality comes first. One can then prove that equality is the only reflexive, symmetric and antisymmetric relation. Hope that helps. Sam Staton 14:49, 15 January 2007 (UTC)

[edit] question

suppose R is a subset of A x A where A = {a,b} R={(a,a),(b,b)} is this antisymmetric? if yes then why?

The article gives "Greater than or equal to" as an example of antisimmetry. So, the "equal to" "a=b" relation here is just a particular case of it. Futhermore, the article clearly states that the equality relationship is both simmetric and antisimmetric ;) Actually, any xRy=true coincides with xRx, which satisfies both yRx and x=y. These two implications are the only requrements for the antidependency. So, any of your relations are antisimmetric. Javalenok'
Am I wrong in saying that.. you're wrong? Greater than or equal to isn't antisymmetric, right? I'm pretty sure its neither symmetric nor antisymmetric. Fresheneesz 04:47, 13 December 2005 (UTC)
No, as the article says, the relation "greator than or equal to" is antisymmetric. Paul August ☎ 05:32, 13 December 2005 (UTC)

Uh, .. where? This article? but greater than or equal to has cases where aRb exists but a does not = b... so .. i'm pretty sure i'm right. 67.161.46.169 02:09, 14 December 2005 (UTC)

The fourth line of the article says: Inequalities are antisymmetric, since for different numbers a and b not both a ≤ b and a ≥ b can be true. In other words if both a ≤ b and a ≥ b, then a = b. That is the definition of antisymmetric. Paul August ☎ 03:03, 14 December 2005 (UTC)

[edit] Picture

I had a picture of the equality relation, Arthur Rubin deleted it. Why? First off, we need examples of antisymmetric relations. Secondly, pictures most definately do illustrate the concept. Please tell me why my picture should not be edited back in. Fresheneesz 02:32, 14 December 2005 (UTC)

Your picture incorrectly said that for x ≠ y, xRy "must be false". Paul August ☎ 03:03, 14 December 2005 (UTC)
Oh ok yea you're right. Needs a better picture then. Crap.. my picture should be deleted in that case. How do I do that? Fresheneesz 03:10, 14 December 2005 (UTC)

[edit] Table

The table under examples is IMHO very confusing. The relation "x is even y is odd" is antisymmetric, but I don't understand how the table helps illustrating it. It seems that the table tries to illustrate some conditions for an unspecified relation to be antisymmetric but it is very involved and hardly helps understanding. Can it be removed?

Perhaps it would be helpful to instead have an example of something that's not antisymmetric. E.g., a pre-order like "is cheaper than" for fruits? (if two different fruits have the same price, then it is not anti-symmetric). 130.238.11.101 17:23, 16 February 2006 (UTC)