# Definition talk:Relation

Just noted that relation on $S$ is defined under 'Relation on a Set' as well as under 'Endorelation'. --Lord_Farin 09:36, 20 January 2012 (EST)

N'est-ce pas? My view is that the "Endorelation" definition should stand (with its "also known as a relation on a set" kept), and the first instance of "Relation on a set" should be removed from here and put into Definition:Left-Total Relation instead with a similar "also known as", and to both of them a warning should be added that "relation on a set" has an alternative meaning, is ambiguous and therefore discouraged. --prime mover 10:26, 20 January 2012 (EST)
Agreed, that's best. Although my mind says that a relation on a set is an endorelation, maximal rigour and clarity should be achieved, so this is the only viable option. --Lord_Farin 10:47, 20 January 2012 (EST)
Job done. Does that work for you? --prime mover 13:34, 20 January 2012 (EST)
Sure. One down, hundreds to go. But we are undeterred. --Lord_Farin 18:20, 20 January 2012 (EST)

## Question

Hmm, there seems to be a problem here. It doesn't make sense to ask if a set $R \subseteq X \times Y$ is left-total or right-total or neither, because neither $X$ nor $Y$ is uniquely determined by $R$. The only sensible thing is to ask if the ordered triple $\left({X, Y, R}\right)$ is "left-total" or "right-total." So either we're going to have to say that $\left({X, Y, R}\right)$ is the relation, or we're going to have to call $\left({X, Y, R}\right)$ a [name] and then say that the [name] is "left-total" or "right-total" or neither. --abcxyz (talk) 17:56, 20 October 2012 (UTC)

I think when someone asks if a relation is left-total or right-total the underlying set of the relational structure is implied. --Jshflynn (talk) 18:17, 20 October 2012 (UTC)
what he said --prime mover (talk) 18:18, 20 October 2012 (UTC)
So I guess you're saying it's something like Definition:Interior (Topology) where we say "interior" instead of the more specific "$\vartheta$-interior." Thanks; your comment just made me think of that. For some reason, it didn't occur to me. Fine, then. --abcxyz (talk) 18:30, 20 October 2012 (UTC)
I just realised I'm technically wrong because a relational structure only works for endorelations. For notationphiles have a look at Bourbaki's Theory of Sets Chapter 2 Section 3.1 to see how they ordered the triple that abcxyz describes differently :) --Jshflynn (talk) 18:28, 20 October 2012 (UTC)
The definition of which you refer, abcxyz, is already in this page at Definition:Relation#Relation as an Ordered Pair, so we already have the required base covered. I can see where you are coming from: it can be argued that by just defining "relation" as just the subset of the cartesian product, it could be construed that the relation is just that. I suppose it is worth labouring the point - or would you suggest that the section alluded to (i.e. the one that treats a relation as a pair consisting of the product and the subset of that product) be made the main section?
Also note that the page Equality of Relations is explicit in its inclusion of the domain and range in the things it is defining equality on.
And the page defining left-total and right-total themselves include the domain and range in the discussion, by way of defining what they are. --prime mover (talk) 19:25, 20 October 2012 (UTC)
I just read all of this, and the first thing coming to my mind is: "It does make sense. The only problem is that we're not sure why." A similar reasoning applies to Talk:Open Set may not be Open Ball. We could of course write every thing with so much rigour that a proof checker would approve of it (with sufficient time input). However, this would inevitably sacrifice the readability and hence the usefulness of the site; I'd rather not go further down that road. Implicit stuff is good to be pointed out on definition pages, but critical is that it is also stressed that, when no risk of ambiguity (that's a very loose concept, of course - nonetheless) arises, we simplify notation for readability and brevity. --Lord_Farin (talk) 19:26, 20 October 2012 (UTC)
I don't think that defining a relation as $\mathcal R = \left({X, Y, R}\right)$, where $R \subseteq X \times Y$, is even close to unreadable. If we keep it as it is, I'm concerned that someone new to these concepts might look at Definition:Relation, Definition:Set Equality, and Equality of Relations and see them as inconsistent. Obviously, I prefer defining a relation as $\mathcal R = \left({X, Y, R}\right)$, but is it worth changing or does it not really add any clarity? --abcxyz (talk) 20:54, 20 October 2012 (UTC)
As I said, it's already covered in Definition:Relation#Relation as an Ordered Pair. It's not unreadable, it's just unwieldy. --prime mover (talk) 22:16, 20 October 2012 (UTC)
Why write everything once? Lord_Farin I need you to implement a scroll bar on every page such that scrolling toward the right you have everything as explicit as metamath and toward the left you have it as terse and efficient as specialists write to one another. Oh and on top of that can you create some PHP script working with the regular expressions of Latex so that when I hover over some symbol it links me to the correct page? All by Monday please :) --Jshflynn (talk) 19:45, 20 October 2012 (UTC)

Fortunately, I have been working on that recently on my super secret PW clone. These are interesting ideas, though (but many specialists' notes would be no more than "Trivial", "straightforward", "obvious" and their consorts, of course. --Lord_Farin (talk) 19:55, 20 October 2012 (UTC)

Btw, would you be okay with the hovering thingie implemented in JavaScript (that's better suited to MathJax)? :) --Lord_Farin (talk) 19:57, 20 October 2012 (UTC)

On a more serious note, people interested can discuss stuff at http://mathim.com/ProofWiki. --Lord_Farin (talk) 19:59, 20 October 2012 (UTC)

## As Ordered Triple

As the new section Definition:Relation#Relation as an Ordered Triple is effectively a weaker implementation of Definition:Relation#Relation as an Ordered Pair, and does not add anything, I wonder whether its presence is viable. Thoughts? --prime mover (talk) 22:52, 20 October 2012 (UTC)

I added it because the cases where $S = \varnothing$ or $T = \varnothing$ work out differently. --abcxyz (talk) 22:55, 20 October 2012 (UTC)
Fair enough. Now what we have to do is write a page explaining the differences between all three different approaches, and why it's important and what the consequences are of failing to take into account all these details. I'll leave it to you to fill in the details.
Also, a citation for your ordered triple approach would be nice. Unless you invented it, that is. --prime mover (talk) 23:08, 20 October 2012 (UTC)
"Write a page explaining the differences...": Well, I put this in Definition:Relation#Note, but you apparently don't like it (if I am correct in interpreting "trash can" as "out of the page"). I'm sorry if the passage sounds biased, but I couldn't really come up with anything better then. Of course, as always, I am open to suggestions. --abcxyz (talk) 16:23, 11 January 2013 (UTC)

Abcxyz I like your triple notation. See if you can find a much more interesting problem than this one.

$\left({X_1,Y_1,\mathcal R_1} \right) \left( {\supseteq, \subseteq, \subseteq} \right) \left({X_2,Y_2,\mathcal R_2} \right)$

(Pointwise) Implies:

$\left({X_1,Y_1,\mathcal R_1} \right) \left( {\cup, \cap, \leftarrow} \right) \left({X_2,Y_2,\mathcal R_2} \right) = \left({X_1,Y_1,\mathcal R_1} \right)$

Where $\left({X_1,Y_1,\mathcal R_1} \right) \left( {\cup, \cap, \leftarrow} \right) \left({X_2,Y_2,\mathcal R_2} \right)$ just means $\left({X_1 \cup X_2,Y_1 \cap Y_2,\mathcal R_1 \cap \left( {\left({X_1 \cup X_2} \right) \times \left({Y_1 \cap Y_2} \right)} \right)} \right)$. --Jshflynn (talk) 21:12, 22 October 2012 (UTC)

I don't understand what you mean. What's the problem? Could you please explain? --abcxyz (talk) 18:58, 23 October 2012 (UTC)
Give it no further thought abcxyz. I don't think I was thinking straight at the time. --Jshflynn (talk) 17:46, 31 October 2012 (UTC)

## Terminology

We currently call a subset of $S \times T$ a "relation on $S \times T$". That is, we actually define "relation on" to mean "subset of", but only when applied to products. This seems a bit odd to me. In the two-set case, I would think terminology like "relation from $S$ to $T$" or "relation between $S$ and $T$" might be appropriate. As for the general case, where do people tend to discuss relations among elements of infinitely many sets as relations? Doesn't it make sense to just switch entirely to set-theoretic terminology at some point, unless perhaps you're doing some weird set-theoretic thing that will require a completely different definition of relation anyway?--Dfeuer (talk) 18:33, 10 January 2013 (UTC)

What's your point? --prime mover (talk) 19:09, 10 January 2013 (UTC)
That we might want to consider different terminology on this page. I'll look around and try to find some sources. I may even buy some.--Dfeuer (talk) 22:43, 10 January 2013 (UTC)
There is some discussion of the matter in Devlin's "The Joy of Sets" which I still need to study in depth (I'm currently doing other things). In general I would consider limiting any changes made on the basis of the philosophy of the above paragraph to the "also see" or "also defined as" or "also known as" sections. Major amendments of existing work purely on the basis of a particular angle from which you can see something is unhelpful because it may have adverse effects on pages which link to, and are dependent upon, specific definitions with specific terminology.
Bottom line is: there is shitloads of material on this site which is all more-or-less established, and the more fundamental the concepts the more well-established it is. We appreciate that there are alternative ways of viewing stuff (in fact most stuff has multiple ways to approach it). But we do not want random people turning up and just amending the first page they see, just for the sake of changing stuff to suit their own solipsistic worldview.
To that end: this page is one of those. It's a piece of cake to come in and just change stuff that's already there because you think it's rubbish or wrong - ensuring that the entire site hangs together as a piece of self-consistent architecture is something that we have been working on for half a decade.
I appreciate that this might sound like "authority by seniority" - well get this, it's exactly what it is. --prime mover (talk) 22:55, 10 January 2013 (UTC)
Also, please do note that the general case covers finitary relations only; nonetheless, developments of logic exist admitting infinitary operations. --Lord_Farin (talk) 22:58, 10 January 2013 (UTC)

The way it uses language formally is annoying. Specifically, "$R$ is a relation on $S$" means "$R$ is a subset of $S$ (and I promise that S is a product of sets)", while "$R$ is an endorelation on $S$" means "$R$ is a subset of $S \times S$. Although an endorelation is a kind of relation, and the language is parallel, you can't substitute one for the other (but we do it anyway because it's what we're used to). --Dfeuer (talk) 23:20, 10 January 2013 (UTC)

## Reappraisal of approach

As ever, I have (finally) deferred to better minds than mine, and amended the definition of "relation" specifically to be the "ordered triple" approach, as this appears to be the most rigorous approach of the lot. Thanks (belated) to abcxyz, DFeuer (in particular) for championing their corner, apologies for being boneheaded enough not to want to listen.

There's plenty of work ahead to make sure the site is internally consistent with this. This will not be an immediately accomplished task. If you see where the approach taken does not match the "ordered triple" approach, please flag it up as a "needs maintenance" issue.

I know about the definition of "function", I will get to this in due course. --prime mover (talk) 12:06, 8 September 2016 (EDT)

## Relations for Classes

The page says that you can use this for both sets and classes, but the definition of a triple (and hence the definition of an ordered pair) involves having these objects as elements of something, and by the definition of a class you can't generally have classes as elements because of proper classes. --HumblePi (talk) 19:27, 18 April 2017 (EDT)

I am uncomfortable about that statement myself. We have had a few people decide to change some of the pages in various random places on $\mathsf{Pr} \infty \mathsf{fWiki}$ in the past, without adopting a comprehensive and consistent approach to class tbeory. It is not something I have studied in any detail myself, except for having read the opening chapters of Bernays, so I don't feel able to attack the area myself.
I do believe that just adding "or classes" to existing pages is the correct approach, but I have not done anything about fixing it for existing pages for fear of provoking hissy fits.
I would very much like for existing pages which have evolved from the initial more-or-less-carefully crafted treatment based on ZFC to stand as they are (or were originally conceived), and for there to be a subpage to any of these set-theoretical pages to be added expressing the necessary words to expand the concept to classes. But, as I say, I have not yet formulated the details of how this ought to be done. --prime mover (talk) 01:23, 19 April 2017 (EDT)