Talk:Converse relation

Notes & Queries
Jon Awbrey 03:25, 3 February 2006 (UTC)

In the Relation algebra community this operation is invariably called converse. What's an example of a community that more commonly refers to it as inverse? --Vaughan Pratt (talk) 19:31, 18 January 2010 (UTC)
 * I don't know if it's a matter of field rather than author's preferences. It's clearly widely used though. Pcap ping  03:55, 19 January 2010 (UTC)

Is it worth mentioning that if L
$$L \circ L^{-1} \neq \mathrm{id}$$ in general; in the case of equality it is in fact true that $$L$$ and $$L^{-1}$$ are functions which are mutually inverse. Is this worth mentioning?

--Daviddwd (talk) 19:41, 14 September 2014 (UTC) Perhaps then we really should call this the 'opposite relation' because that is what is naively expected, and also because the dagger functor on the category of relations assigning to each relation the opposite relation (in the opposite category) assigns with what is called here the 'inverse relation'. — Preceding unsigned comment added by Daviddwd (talk • contribs) 20:37, 14 September 2014 (UTC)

Please make this article easier to understand by non-mathematicians!
I have a PhD in Computer Science, and I've certainly had reasonable exposure to math and logic over the years, but I find this article unbearably difficult to unpack. The main problem is the high density of jargon and the domain-specific notation. Obviously the reason for that is precision, which is important also. But the problem is that a definition of an unfamiliar term is often given in terms of 2-3 other unfamiliar terms, which in turn are defined in terms of more unfamiliar terms, etc.. I find myself diving through several layers of dependency definitions -- each with a different wikipedia page -- just to unpack one definition. And the problem with specialized notation is that it is not even clear how to look up the definition of a symbol such as "*" or $$\in$$ (which I do know, but I'm just using as an example). Ask yourself: What would a reader type into a google search, to find out what the $$\in$$ symbol means? Examples help a lot. And plain language definitions help enormously, even if they are imprecise, though of course they should be clearly noted as being imprecise. The key point in a plain language definition is to avoid domain-specific jargon. For example: "Roughly speaking, a foo is . . . ". And later: "More precisely, a foo is . . . " (with full rigor and jargon). As a case in point, I was just looking up the definition of "inverse relation" on wikipedia, and the explanation talked about a "semigroup with involution", so I had to look up that, which was defined in terms of a "semigroup", so I had to look up that page, which says that "A semigroup generalizes a monoid", so I had to look up "monoid". . . except that I gave up at that point. :( (Stack overflow?) There are two main use cases that a page like this should address, and they are different: (a) someone runs across an unfamiliar term and reads the page to get a rough idea of what that term means; and (b) someone wants to dig deeply and precisely into the meaning of the term.  The (possibly imprecise) plain language definition should be given first, free of jargon.  The gory details and jargon should come later.  I do think it is important to introduce the jargon that is used in the field, but it should be fairly clearly separated from first providing a layman's (approximate) definition, so that readers can get the gist of what the term is about before they face the prospect of a deeply nested recursive traversal through many pages of jargon-filled definitions. I hope the above suggestions are helpful and don't just sound like complaints. I know it is hard to write such things in widely understandable ways, and I very much appreciate the efforts of all editors who contribute. Thanks! -- DBooth (talk) 16:38, 17 April 2015 (UTC)
 * Yeah this page suffers from some bad structure and writing. But I think it's unreasonable to explain set-builder notation in every article, other than by linking to it. See write one level down. 86.127.138.67 (talk) 04:54, 18 April 2015 (UTC)
 * Also this article starts [and in fact did so even when you tagged it] with a exactly the [somewhat imprecise but] plain English description followed by an example in exactly the same plain English. These are the first two sentences in the article. So what exactly are you complaining about? You can't have an imprecise English sentence translating everything in the article. Since this is actually a short article, instead of writing (in fact copy-pasting) these longs rants on talk pages, why don't you add your favorite plain English explanations to whatever bothers you as too formal? 86.127.138.67 (talk) 04:14, 19 April 2015 (UTC)

Mention that inverse relation is involution in WP:LEAD or not?
In relation to the discussion above, on one hand it's probably important for the lead to warn/say that the inverse relation is not a group inverse and that it is an involution instead. On the other hand, it's not possible to cram all the necessary definitions in the lead to fully explain what this means. 86.127.138.67 (talk) 05:50, 18 April 2015 (UTC)

Transposed or Converse
As comments above indicate, Inverse relation is not appropriate as a title for this article which should be named Transpose relation or Converse relation. True, some early authors use Inverse relation, but propositions at heterogeneous relation show that, only under certain conditions, is the identity relation contained in the product of a relation and its transpose. Furthermore, on page 79 of Relations and Graphs by Schmidt and Strohlein, these comments precede an exercise:
 * 4.4.2 A relation R is called right-invertible if there exists a relation X with RX = I, and left-invertible if there exists a Y with YR = I. X and Y are then called the right and left inverse of R, respectively. Right- and left-invertible relations are called invertible. For invertible homogeneous relations all right and left inverses coincide; the notion inverse R–1 is used. Then R–1 = RT holds.

Comments are invited; necessity for the Move seems apparent. — Rgdboer (talk) 22:06, 18 June 2018 (UTC)

Move done. — Rgdboer (talk) 22:28, 19 June 2018 (UTC)


 * Gee. A whole day for comments before this poor idea of a move was carried out. --Bill Cherowitzo (talk) 05:58, 21 June 2018 (UTC)

And why is it a "poor idea"? — Rgdboer (talk) 22:50, 21 June 2018 (UTC)

No reply! (Note the 2010 contribution above by Vaughn Pratt calling for Converse relation.)

Moving on: Completing a Move involves using "What links here?" to clarify article name-change. The redirect inversely related was found pointing here. No wonder the article was considered part of Project Statistics (negative correlation expresses inverse random relations) until June 19! Currently 48 editors Watch this article, but only 8 have reviewed recent changes. — Rgdboer (talk) 22:19, 25 June 2018 (UTC)


 * Even if "inverse" for $$R^{-1}$$ may be confused with $$1/x$$, the term is widely used in all math domains (e..g. when $$R$$ is a function, or in linear algebra), and must be given as a synonym of "converse". More generally, it is not the goal of Wikipedia to establish the "best" or "logical" nomenclature, or best approach to a subject. Like any good dictionary, Wikiepdia should record any widely used names, no matter how ambiguous or illogical. E.g., when explaining "linear function", Wikipedia must note that it may mean $$A x + B$$, even though that is more properly called "affine".--Jorge Stolfi (talk) 22:19, 15 November 2023 (UTC)

Ambiguous direct quote
I recently modified, in part, to read: Relations that are both right- and left-invertible are called invertible. However, I had failed to notice that this was a direct quote from a reference, where the corresponding passage read: Right- and left-invertible relations are called invertible. User:Rgdboer, quite rightly, reverted my mistaken edit, for which I've thanked them.

However, a problem remains, to wit: the quoted text is ambiguous, and could easily be miscontrued to mean: "a right-invertible relation is called invertible, and a left-invertible relation is called invertible". I believe that the way I expressed the definition above is both correct and unambiguous – but I don't have a better source to hand than the ambiguous one presently in use. When time permits, I'll seek one to use here. yoyo (talk) 05:01, 3 August 2018 (UTC)


 * I agree that your version is more clear, and suggest to do without direct quoting here. - Jochen Burghardt (talk) 12:16, 3 August 2018 (UTC)


 * Standing alone, the sentence is ambiguous. Reading the whole quotation, the ambiguity disappears. Right-invertible and left-invertible are defined separately. Though subject to improvement, the sentence builds on the previous definitions and loses its imprecision. As seen in the previous section, this quotation from Gunther Schmidt has been recited to rebut the previous title Inverse relation, which naturally still redirects to this article. — Rgdboer (talk) 21:27, 3 August 2018 (UTC)


 * How does the ambiguity disappear when reading the whole quotation? I find it quite natural to read the quotation to mean that right-invertible relations and left-invertible relations are both called invertible relations for short. But of course, I am not a native speaker. – Tea2min (talk) 05:55, 4 August 2018 (UTC)


 * There are two sentences before the one cited by yoyo: "A relation R is called right-invertible if there exists a relation X with RX = I, and left-invertible if there exists a Y with YR = I. X and Y are then called the right and left inverse of R, respectively. Right- and left-invertible relations are called invertible." The first sentence defines the left and right properties. The second provides labels for X and Y satisfying the properties. The third sentence is supposed ambiguous by yoyo, but with the lead-in by sentences one and two there are two distinct labels. The interpretation of sentence three as fusing the two already named properties sounds unreasonable. — Rgdboer (talk) 22:33, 4 August 2018 (UTC)

Range of membership
The relation of set membership is instructive as an illustration of converse. Writing $$x \in A$$ indicates a relation true when x is in the set A. If A is the range of the relation then the statement is always true, so presumably $$A \subset U$$ for a larger universe U. Thus the range of the set membership relation is the power set of U, written P(U). Set membership is then a subset of U x P(U). Its converse flips these factors.

This observation was put in the article 19 November 2018 and reverted 2 April 2022. Discussion is invited. Rgdboer (talk) 04:00, 2 April 2022 (UTC) Rgdboer (talk) 04:07, 2 April 2022 (UTC) Rgdboer (talk) 04:09, 2 April 2022 (UTC)


 * I was about to revert the revert, and fix the text about domain/range. But then I thought, $$(\ni) = (\in)^T$$ is not a particularly interesting example, and $$\ni$$ is very rarely used. I suggest to add the example $$ (<) = (>)^T $$ instead, which is (boring but) well-known from elementary mathematics. - Jochen Burghardt (talk) 17:36, 2 April 2022 (UTC)

Okay. The thought provoking aspect of membership converse may be better discussed in Universe (mathematics) or in the Set membership article itself. The homogeneous relations don’t bring up the non-commutative aspect of converse on domain/range that makes Set membership different.Rgdboer (talk) 00:01, 3 April 2022 (UTC)

See Element (mathematics) for presentation elsewhere.Rgdboer (talk) 01:22, 3 April 2022 (UTC)

Article now shows $$A \ni \in B \Leftrightarrow A \cap B \ne \empty.$$ Rgdboer (talk) 04:23, 6 April 2022 (UTC)

"Inverse" is a much more common name than "converse"
The renaming of the article from "inverse" to "converse" was hurried and unjustified. Besides being more logical and intuitive, according to Google the phrase "inverse of a relation" is at least 5 times more common than "converse of a relation. (And indeed, even though I have used the concept for many years, I never seen it named "converse", or "transpose". I suppose that the "common" name depends on which sub-area of mathematics one lives in.  Maybe "converse" is more common among logicians, by influence of (or specifically for) the "$$\to$$" logical connective?) While it may not be worth reversing the move, "inverse" should definitely be listed as a synonym on the head paragraph, at least at the same level as "converse". Jorge Stolfi (talk) 15:37, 18 November 2023 (UTC)
 * Currently inverse relation is a disambiguation page with four alternatives. One is inversely related or "negative relationship", meaning a correlation coefficient in the negative range. Evidently the term inverse is broadly used in logic and mathematics: see Inverse (disambiguation). Some inflation of Google numbers can be expected due to promoters of "inverse function"; but bijection covers the topic. — Rgdboer (talk) 00:45, 19 November 2023 (UTC)
 * Who would be those "promoters of 'inverse function'"??? The phrase "inverse of a relation" (with quotes, therefore not counting "inverse of a function") has over 450'000 hits, and is used in many textbooks. Besides, a function is a special case of relation, and the inverse of a bijective function is precisely the same thing as its inverse as a relation -- and it is always called "inverse", never the converse or transpose.  And a non-injective function has an inverse too (which, when mentioned, is called by this name), namely its inverse as a relation --- except that, in this case, the inverse is not a function, just a relation. The fact is that the most common name for this concept is "inverse", not "converse".   Wikipedia articles should be named with the most common name for the concept, irrespective of whether it is confusing, illogical, or frowned upon by some "authority". And that name must definitely be given in the lead. All the best, --Jorge Stolfi (talk) 13:36, 19 November 2023 (UTC)
 * Every relation has a converse, and every function is a relation. Thus, every function has a converse. Now if inverse is used instead of converse, then one is led to "every function has an inverse" which is not tenable. Inverse functions exist only conditionally. The common practice, indicated by usage statistics, sets up the linguistic trap. Prevention of misinformation is good policy. — Rgdboer (talk) 01:18, 20 November 2023 (UTC)
 * Every function does have an inverse. The inverse of a function is not always a function, but it is a relation.  So, if $$f$$ is a function, one may not be allowed to write $$f^{-1}(y)$$ if $$y$$ is an element of the image set, and expect it it to be an element of the domain (because that notation is defined only for funtions); but one can write $$f^{-1}(X)$$ if $$X$$ is any set, and get a subset of the domain -- just as one can do with any relation. But the function issue is not the point. The main point is that, when talking about relations, the concept that you call "converse" is much more commonly called "inverse" -- by a very wide margin, and by totally respectable sources.  Therefore, that name should be used in Wikipedia -- irrespective of whether some consider it misleading or improper.   Misinformation is omitting "inverse" and pretending that the common name is "converse". The mission of Wikipedia is to record current nomenclature and explain the meaning of terms as readers may encounter them -- not to push for "better" nomenclature. All the best, --Jorge Stolfi (talk) 12:51, 20 November 2023 (UTC)
 * Reliable sources are the standard of reference here. Sources such as now cited for inverse are not reliable, given the indicated trap. Gunther Schmidt is a reliable source for relations and his book Relational Topology (2018) uses the terms conversion and transposition for obtaining this relation from an original. Misleading sources are not reliable. — Rgdboer (talk) 00:23, 21 November 2023 (UTC)
 * A "reliable source" is a source that can be trusted to get a fact right. A NIST website is a more reliable source for the density of gold than a high-school chemistry textbook. The CIA World Factbook is a more reliable source for the area of Australia than some Ausralian newspaper. An "authoritative source" is a source that is expected to give the official text or sense of a decision or definition. The US Patent Office website is an authoritative source for the date and status of a US patent; Time Magazine is not. The 9th BIPM Bulletin is an authoritative source for the definition of "mole", sice the BIPM is the entity that defines the mole; a chemistry book, no matter how respected, is not. A "correct source" is a source that has a correct formula or proof of a mathematical fact. But there is NO such thing as a "reliable", "authoritative", or "correct" source for mathematical nomenclature.  There is only common usage. And the most common name for this concept, by far, is "inverse of a relation"; "converse" is a minority use.  You can check that yourself by looking for recent books, e. g. this one. I insist: Wikipedia articles must not try to lead or persuade people to use what one believe to be a "correct" or "proper" nomenclature. THAT IS WRONG.  We can and should do that in our own books, but Wikipedia is NOT for that. Like any good dictionary, it should record common usage, no matter how illogical, confusing, or awkward, no matter what textbooks and "authorities" say one "should" use. And again, there is no trap.  The use of "inverse" for relations is completely consistent with its use for functions.  It is in fact precisely the same concept. All the best, --Jorge Stolfi (talk) 18:00, 22 November 2023 (UTC)
 * The sources given for converse are all notable authors who promulgate the calculus of relations as experts. Writers using inverse are apparently dilettantes unaware of practice among the knowledgeable. The special section of the article on inverse of a function notes the important fact that the inverse of a function may not be a function. Expecting a general reader to distinguish between "inverse of a function" and "inverse function" seems unreasonable. The dangerous language is perfectly avoidable by comparing sources according to standards. — Rgdboer (talk) 02:18, 24 November 2023 (UTC)