Talk:Transitive relation

message: I erased a couple lines i consider wrong: they say that, for a transitive relation, it's equivalent to be irreflexive or asymmetric. In other words, "transitive and irreflexive" if and only if "transitive and asymmetric". This should be false: consider a relation R over {a,b} where aRa, aRb, bRb (and b is not in relation with a). This is transitive and asymmetric, but not irreflexive. Please recover those lines if i'm wrong.
panurge.
:A bit late responding, but this is incorrect. This relation is antisymmetric, not asymmetric. All asymmetric relations are necessarily irreflexive (the converse is not necessarily true, but is true in the case where the relation is transitive). This may be a difference in usage. I found a reference for this and will add it. Dcoetzee 06:53, 20 August 2013 (UTC)
thread_title: Erased a couple lines

message: (1,1) ^ (1,2) => (1,2) is this transitive? or x != y != z?
:What I think you are asking is: "does the definition of transitivity requires the elements to be distinct?". The answer is negative. For exemple, if you have only one element "a", then you can define only two relations ("aRa", and "not(aRa)"). They both are transitive. You may also check that, among the 16 possible relations over two elements, 13 are transitive. You simply don't care about the elements being distinct, just apply the definition. So your exemple is a totally legal application of that definition. Panurge
However, if there are less than 3 elements, then reflexivity must hold for all a in the set, and symmetry must hold for all a, b in the set for transitivity to hold.  — Preceding unsigned comment added by 128.12.255.132 (talk) 23:41, 15 October 2016 (UTC)
thread_title: Are the element required to be distinct?

message: *That starting sentence could really use some work. Someone coming in to learn about this would probably not understand what this property is. ---You'reMyJuliet 19:55, 3 June 2006 (UTC)
thread_title: Starting sentence

message: There's no need for a logical formula in the lede. It will only serve to confuse naive readers, since anyone who is ready to read the predicate logic formula will already know what transitivity is.  The English definition is enough of a definition. I would prefer to see an example in the lede, but when I added one it was reverted. — Carl (CBM · talk) 22:36, 12 July 2008 (UTC)
replies: text: I wasn't sure myself; my reason for leaving it in was consistency and, to some extent, advertising of predicate logic.  I continue to be surprised at the number of people who need concepts such as transitivity in their work or studies but largely fail to apply them correctly because of a lack of basic training in predicate logic. You can't really understand transitivity unless you understand predicate logic. Rp (talk) 13:44, 13 July 2008 (UTC)
replies: text: I don't think putting a formula in the lede will persuade people who don't understand predicate logic that they should go learn it — it will merely discourage them from reading this article as well. —David Eppstein (talk) 14:57, 13 July 2008 (UTC)

text: I disagree. "Factual" is not the final criterion for including something in an article. I don't think that there's any actual benefit to including a predicate calculus formula in the article. Also, you have been re-inserting some nonsense text at the very beginning of the article. — Carl (CBM · talk) 02:00, 7 December 2008 (UTC)

text: No, I mean the "insert math formula here" at the very beginning of the article. — Carl (CBM · talk) 02:09, 7 December 2008 (UTC)

text: OK, thanks for fixing that. Here's the issue with the formula. First, it isn't integrated in the article at all. It used to be that there was a formula in the article (hence my post above) but I removed it. The English sentence at the beginning of the article is perfectly lear and rigorous. Any gain from the symbolic formula is an illusion. — Carl (CBM · talk) 02:17, 7 December 2008 (UTC)

text: "No harm" is, like I was saying, not the standard we use for inclusion of material. Unless someone else supports the inclusion, I'll remove the formula (again) in a day or two. Please be aware that we prefer to make content decisions by discussion - repeatedly making the same edit when other people don't agree with it is considered bad form and may lead to you being blocked from editing. — Carl (CBM · talk) 03:10, 7 December 2008 (UTC)
replies: text: I agree with Carl and David: the article appears perfectly rigorous in its present form, and more legible than it was with formulas. Rp (talk) 07:55, 14 January 2009 (UTC)

text: A hint: if you're looking for Godwin's law, it should be under rhetoric, not mathematical logic. —David Eppstein (talk) 04:06, 7 December 2008 (UTC)
thread_title:  "predicate logic"formula 

message: Maybe the two can be merged? 
Lone wolfII (talk) 12:15, 21 March 2010 (UTC)
replies: text: Especially since they have had contradictory examples; the first reports that matrilineal ancestor is transitive, and the second has reported that "ancestor" is not transitive.  — Arthur Rubin  (talk) 16:27, 31 March 2010 (UTC)
thread_title:  Two Example Sections 

message: Does everyone believe the mother example is clear/strong enough? I think while it is technically correct it isn't the greatest example:
"if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire."
Alice is the grandmother of Claire while not the maternal mother Alice is still a mother of Claire. I think there are way better examples of non-transitive relations that don't have this grey area. — Preceding unsigned comment added by Timlempicki (talk • contribs) 
:Are you possibly a non-native English speaker? Because I can not imagine anyone who is one claiming that a grandmother is a type of a mother, or that "maternal mother" is a useful distinction to make versus other mothers. —David Eppstein (talk) 00:34, 12 February 2016 (UTC)
thread_title: Example Section

message: Explaining/linking set (mathematics) and ordered pair is not the problem. As far as I understand the setting, you have a set S and want to count the number of transitive relations on S under some restriction.
A relation R on S is transitive if 
:∀x,y,z∈S. (x,y)∈R ∧ (y,z)∈R ⇒ (x,z)∈R.
I repeat the definition here since in your JAMC paper (p.1-->247) you use a different one, viz. 
:∀x,y,z∈S. (x,y)∈S ∧ (y,z)∈S ⇒ (x,z)∈S.
Both definitions are not equivalent, and R does not appear at all in your definition of "R is transitive".
Consequently, S is usually not a set of pairs. I guess you mean "R contains exactly two ordered pairs", i.e. you count the number of all transitive relations R on S such that R has exactly two elements. This would be compatible with your Definition 1 in the JAMC paper. Please comment and help me to understand your intentions. Many thanks in advance. Best regards - Jochen Burghardt (talk) 19:48, 9 December 2021 (UTC)
thread_title:  JAMC paper