Talk:Transitive relation

Erased a couple lines

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)

Are the element required to be distinct?

(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)

Starting sentence

• 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)

Agreed. It's too vague. Transitivity is not just three things being related, but two identical relationships between three elements. i.e. if A is related to B and B is related to C in the same manner, then A is also related to C in the same manner, typically expressed by an operator. For example, if A is a subset of B and B is a subset of C, then A is a subset of C.199.243.165.82 (talk) 16:29, 24 August 2009 (UTC)

"predicate logic"formula

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)

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)

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)

This is all incredibly irrelevant it deserves to go on the page whether or not it was dissuade the entire universe from reading it ;if its rel;evant and factual leave it —Preceding unsigned comment added by 128.243.253.113 (talk) 01:56, 7 December 2008 (UTC)

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)

'nonsense text'?you mean the QL of transitivity?that isnt nonsense?and if someone wanted to find out how transitivity can be translated in QL then shouldn't we put it on? —Preceding unsigned comment added by 128.243.253.114 (talk) 02:05, 7 December 2008 (UTC)

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

if i did that i didn't mean it —Preceding unsigned comment added by 128.243.253.111 (talk) 02:12, 7 December 2008 (UTC)

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)

I disagree that it is obvious that the first statement implies the formula especially to someone not experienced in the field.it may be that someone comes alongs and whats to know the specific answer to 'what is QL formula for transitivity?' and are unable to work it out for themselves,it aint doing any harm —Preceding unsigned comment added by 128.243.253.114 (talk) 02:26, 7 December 2008 (UTC)

"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)

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)

oh yeh and because what most other people say is right ?just like nazis where everyone said it was ok to kill the jews.it is clearly relevant so i cant possibly see why not —Preceding unsigned comment added by 128.243.253.112 (talk) 04:00, 7 December 2008 (UTC)

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)

I remember that from a while ago.Anyway this is philosophical logic not mathematical. —Preceding unsigned comment added by 128.243.253.113 (talk) 04:26, 7 December 2008 (UTC)

They may teach mathematics but they clearly don't teach etiquette at our university. I mean what do the Nazis have to do with anything? --128.243.253.114 (talk) 06:06, 8 December 2008 (UTC)

what a dumb question,the nazis are associated with many things such as genocide and totalitarianism —Preceding unsigned comment added by 128.243.253.112 (talk) 13:28, 8 December 2008 (UTC)

A dumb question. In light of article, is the "mathematical syntax" formula even correct? Look at the second example in the Simplification section. We certainly do not want that. I may be making some huge mistake here, but I think that there is a reason for most of the definitions that I have seen to use natural language. Could some mathematician weigh in and put my worry to rest? Can we just say "mathematical syntax," or do we have to say exactly which logic for the formula to make sense? — Preceding unsigned comment added by 99.137.50.151 (talk) 05:41, 8 February 2012 (UTC)

Two Example Sections

Maybe the two can be merged? Lone wolfII (talk) 12:15, 21 March 2010 (UTC)

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)

Substitution or Transitivity?

My teacher(who is sometimes mistaken in what he teaches us) says that "if A=B and B=C then A=C" is transitive but "if A=B and C=B then A=C" is not. I wish to disprove this since a statement of equality works both ways. Anyone willing to prove/disprove my logic? Either one would be appreciated. — Preceding unsigned comment added by 199.36.116.81 (talk) 17:46, 6 February 2014 (UTC)

Equality is (normally) symmetric, so, in regard equality, they are the same. I'm not sure what the precise consequences of "If a R b and c R b than a R c" are. If a reflexive relation, or even if the left-total, then it follows that it's an equivalence relation.

Proof

Assume left-total:

$(\forall x)(\exists y)xRy.$

Then

$xRy\land xRy\implies xRx,$

and we have verified reflexivity.

If $xRy$, then

$yRy\land xRy\implies yRx,$

and we have verified symmetry.

Finally, if $xRy$ and $yRz$, then $zRy$, and

$xRy\land zRy\implies xRz,$

and we have verified transitivity. — Arthur Rubin (talk) 16:40, 7 February 2014 (UTC)

Short, vague lead section

Could someone please clarify in the lead section what "transitivity" actually means, preferably with a definition rather than an example? I had a bash, which I'm told was incorrect. Perhaps someone with more knowledge than me could try to explain it properly? Cnilep (talk) 05:17, 23 June 2014 (UTC)

It means "whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c". That is a definition. It is not an example. —David Eppstein (talk) 06:01, 23 June 2014 (UTC)

First of all: why do you insist that the article is perfect as it is and shouldn't be changed? If someone asks, three times, for clarification, that's an indication that you haven't been clear so far. If that someone is a fellow university professor from another discipline, you probably shouldn't assume that you are correct and your interlocutor is just a moron. (You could make that bet; I might just be a moron, but I doubt your odds.)

Second, if you are going to go to the trouble of explaining to me, here plus twice on your talk page, how I should understand this article's prose, wouldn't it have been more profitable to spend that time editing the article? Make your simplified explanation to all future readers of the article, not just me. It's not about me: it's about improving the project.

Third, do you have some problem with me personally? That I made a general plea to all article contributors here, on the article talk page, and did not address your reply to me on your talk page is an indication that my concern relates to this article. Again, it's not about me. Even if I am a moron, maybe we could try to write an encyclopedia that everyone can understand. Cnilep (talk) 07:46, 23 June 2014 (UTC)

I definitely don't think the article is perfect, only that your edits made it even worse. If you want someone else to tell you which part of the article is the definition instead of me, fine. —David Eppstein (talk) 15:55, 23 June 2014 (UTC)

Example Section

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)

JAMC paper

@Firdous Ahmad Mala: 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)

Yes. I mean what you understood. — Preceding unsigned comment added by Firdous Ahmad Mala (talk • contribs) 19:53, 9 December 2021 (UTC)