Talk:Bijection

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total function

What about total functions? Either a bijective function is also a total function, or the page about total functions is wrong. I suppose it's the former. If so, that should be mentioned here at "Properties". I'd rather have someone write it who is not just supposing things like I am ;) —Preceding unsigned comment added by 80.238.227.222 (talk • contribs) 12:04, July 14, 2006

Unless one is discussion partial functions, every "function" is assumed to be total. Paul August ☎ 15:35, 14 July 2006 (UTC)[reply]

Still, for completeness, the difference between a partial and a total bijection should be explained. ClassA42 (talk) 22:21, 30 March 2008 (UTC)[reply]

I second ClassA42 - the article is not at all clear as to whether bijections are necessarily total. Wootery (talk) 15:10, 6 August 2014 (UTC)[reply]

Too technical

I've changed the lead so that it is no longer too technical for a novice reader (at least IMO), but I haven't really done anything with the rest of the article. Should I do more? Bill Cherowitzo (talk) 17:19, 26 October 2011 (UTC)[reply]

Wondering what other editors think about providing for the very novice reader a couple of examples that may be easily understood. What comes to mind is a planned classroom of 20 students and 20 student desks; the student arrive and each is to sit at a desk; there is one desk for each student. Joefaust (talk) 02:24, 30 October 2011 (UTC)[reply]

Ya some examples would be helpful for this page. As it stands it is still pretty technical.P0PP4B34R732 (talk) 03:01, 30 October 2011 (UTC)[reply]

Totally agree. It also should be made clear that the subject is more about function than something else. 109.206.156.72 (talk) 18:05, 13 October 2017 (UTC)[reply]

Baseball, anyone?

For the sake of readers who might not be familiar with baseball, perhaps another example should be given (instead or in addition). 188.169.229.30 (talk) 02:22, 24 November 2011 (UTC)[reply]

-Seconded :-) I agree, the baseball example was confusing for me as a "non-player" 88.104.128.57 (talk) 20:43, 4 February 2014 (UTC)[reply]

Adding an example using cricket (which, AFAIK, is a sport understood poorly in the few countries which even tolerate such nonsense) is probably the wrong direction... DrKC MD (talk) 03:54, 22 September 2023 (UTC)[reply]

image

When searching google for bijection the image accompanying the wiki text is "a bijection composed of an injection and a surjection" which may be misleading. An image of only a bijection would reduce confusion. If there is a way to enforce which image is grabbed by a google search this change would be good.

First of all, Wikipedia has no control over what Google does. Second, as this article clearly reads in the lead, a bijection is always both an injection and a surjection. What is so misleading?—Anita5192 (talk) 00:13, 22 July 2017 (UTC)[reply]

good job

this is one of the very very few math articles on wiki where the intro is at the right level almost all math articles, the intro is pitched way to high for a general encylopedia congrats to the authors/editors — Preceding unsigned comment added by 50.245.17.105 (talk) 19:14, 18 March 2021 (UTC)[reply]

Is the second sentence of this article correct?

I am not sure that the first sentence (correct one) and the second sentence are equivalent. Could someone correct either me or the page? 96.230.81.2 (talk) 15:56, 31 October 2023 (UTC)[reply]

Fixed. You are correct; thanks. D.Lazard (talk) 16:17, 31 October 2023 (UTC)[reply]

Recent revisions

@Jacobolus: I appreciate your reworking the intro. I'm still unclear on exactly how the notation works that you added; that is, what it would express if spoken out loud. That's due in part to my unfamiliarity with the symbol $\mapsto$ . (Well, now that I see the source, I understand "mapsto".) I assume these expressions are equivalent to something like, "the function X gives Y as a result". Not at all sure if that's correct, though, and I'm sure non-specialists like myself might not understand the precise meaning. This one, $2k-1\mapsto 2k,\,\ldots$ , leaves me completely in the dark. I recognize, of course, that probably very few non-specialist people will seek out this article or subject. Nevertheless, I think improving its accessibility as we're doing is a good exercise in how to similarly improve other specialized articles that might cover more broadly recognized subject areas. So, I have a couple of more suggestions. I would replace the second sentence which contains the notations ("For example...") with the entire second paragraph ("Equivalently, a bijection is an invertible function...."), and move the notation examples lower. Whether using the mapsto symbol or another notation, it would sure help if at least one "spoken" example were given, along the lines of what I wrote above ("function X gives Y", or whatever expression is correct). In most of the article, the simple arrow is used, which I understand to have a different meaning than mapsto. So, maybe mapsto should not be used at all, or if it is used, needs to be defined/explained (briefly) in the text when it is first used. I still think the parenthetical linked "binary relation" could be removed from the first sentence, where I do not believe it is essential and serves primarily as an interruption, and placed lower in the intro, perhaps in a phrase like "also known as binary pairing". DonFB (talk) 06:52, 11 November 2023 (UTC)[reply]

Yeah, ordered pairs notation like

\{(1,2),(3,4),(5,6),\ldots \}

is probably better, or maybe without the curly braces. Let's go back to that version. I moved and added some to the example but I'm not sure that's an improvement. The original idea was to have a very quick example near the top to give an impression of what is meant by "pairing", but if the example gets bigger it gets increasingly distracting. I don't think a "spoken example" would be helpful in the lead section. –jacobolus (t) 15:04, 11 November 2023 (UTC)[reply]

I tried putting "binary relation" as a wikilink on "pairing" but I'm not sure I like it; I think the parenthetical is better. It could be something like "pairing (formally a binary relation)", but making it longer makes this more distracting. The reason to include it is that "pairing" is not a well-defined technical term, and is plausibly ambiguous to people trying to figure out very precisely what a bijection is. Such people (e.g. undergraduates in a first pure mathematics course) are the primary audience of this page, even if it should also ideally be accessible to a broader audience and useful to a more advanced audience. –jacobolus (t) 15:28, 11 November 2023 (UTC)[reply]

I followed up by explicitly mentioning "binary relation" in the following "Definition" section. –jacobolus (t) 00:27, 12 November 2023 (UTC)[reply]

Add: As a matter of principle, I dislike parenthetical expressions in lede sections. If something is important enough to be in the lede—especially the first sentence—it should be expressed directly, rather than as an aside, or "stage whisper". Often, I believe, parentheticals are added by another (smarty pants) editor, not the editor who originally wrote the text. DonFB (talk) 13:41, 11 November 2023 (UTC)[reply]

I don't find your "often" to be reflective of my experience. Parentheticals are common in lead sections because there's a lot of information to cram in and it's hard to do so in a logically structured sentence. Throwing in a parenthetical makes it clear that the additional information is not part of the sentence proper, but is a kind of aside which doesn't break the remaining structure, without being hidden away as it would be in a footnote.

I think you should get over your impression about a stage whisper, which is rarely if ever reflective of Wikipedia authors' intent. –jacobolus (t) 15:31, 11 November 2023 (UTC)[reply]

Thanks for all your changes. The intro is far better and shows it's not necessary to cram a bunch of opaque textbook jargon--forcing readers to chase links deep into the weeds--into the very beginning of a such an article. DonFB (talk) 23:30, 11 November 2023 (UTC)[reply]

Glad I could help. If you see other articles that you find similarly opaque, please don't hesitate to ping me or start a conversation on the math wikiproject or the like. I think there might be still some folks who would prefer to list a function-based definition before a pairing-based definition (definitely more common in typical sources), but you are probably right that the pairing-based version is more accessible, and if we put the function-based versions slightly afterward I think undergraduate math students will still be able to figure out what they need. –jacobolus (t) 23:59, 11 November 2023 (UTC)[reply]

@D.Lazard does this version seem okay to you? –jacobolus (t) 00:56, 12 November 2023 (UTC)[reply]

Jacobolus, just to add a bit more about my thinking:

You wrote:

Parentheticals are common in lead sections because there's a lot of information to cram in and it's hard to do so in a logically structured sentence.

Focusing more on the second part of your comment, "there's a lot of information to cram in", I would say that approach is a significant cause of loading jargon into the first sentence or first paragraph of lede sections. Technically knowledgeable editors tend to want the very early text to be surgically precise, and therefore they use exacting terminology and jargon and add parentheticals if that seems necessary. As a result, they end up writing technically sophisticated ledes that can be all but incomprehensible to typical readers. I've noted that other editors and you have spoken of an intended audience for an article like this--generally, referring to high school or undergrad math students. Here, we may have a real philosophical difference, at least regarding the lede section. My idea of the audience for the lede section of any article is--everyone. I don't think in terms of pitching the lede to any particular segment of the population. I want a lede to start by expressing the fundamental concept in everyday language, without sacrificing basic accuracy, and then build up to more complex concepts. Attempting to immediately "cram in" in a lot of info, in absolutely precise technical language, is the road to opacity. Editors have the remainder of the lede section to begin introducing more complexity, and they have the entire body of the article to dive into the really technical aspects of the subject. Keeping the lede section mostly free of jargon and links from jargon can give the reader an uninterrupted reading experience. That is not a betrayal of an article's "duty" to inform the public. It is, rather, a service to the public. DonFB (talk) 08:57, 12 November 2023 (UTC)[reply]

While this is true, and in general I strongly agree with everything you wrote, in practice the most common group of people reading this article are students in introductory courses. You can see from the page view statistics that there is a strong weekday vs. weekend bias and a strong beginning-of-the-semester vs. typical school holiday bias.

So it's important to make sure that any accessible version of an explanation or definition doesn't promote misconceptions for these students who may be relying on it. –jacobolus (t) 14:06, 12 November 2023 (UTC)[reply]

I disagree with the current version:

The first sentence uses the term "pairing", with a WP:SUBMARINE link to Binary relation, where the term does not appear. Moreover, a reader who needs a definition of this term may search for Pairing, which is unrelated and very technical. This is not only possibly confusing, but also enforce the rather common misconception that using common English words instead of the proper technical words may makes understanding easier.
The example is presented in a much too technical way, since it does not contains anything more than: "adding one" defines a bijection from the even numbers to the odd numbers, and the inverse bijection is "subtracting one". Moreover this example may be confusing, since "adding one" defines also a bijection from the odd numbers to the even numbers, and a bijection from the integers to the integers. So, a much better example would be: For example, "multiplication by two" defines a bijection from the integers to the even numbers, and the inverse bijection is the "division by two".
IMO, starting with the definition as a relation rather than with the definition as a function is not a good idea, since, in practice, bijections are always defined as functions. So readers who have already heard of bijections may be confused by a new point of view, and readers who learn bijections as relations will have, later, to learn another point of view.

D.Lazard (talk) 10:42, 15 November 2023 (UTC)[reply]

I have edited the lead in the line of the above comments. I have also expanded the paragraph on cardinalities. Probably a clean-up is still needed for a more colloquial use of "to", "for", "by", "from", etc. Also, with the new version, it is not needed to know what are the domain and the codomain of a function. D.Lazard (talk) 15:42, 15 November 2023 (UTC)[reply]

I think the lead is more accessible than before @DonFB asked their question (because we e.g. include inline definitions for injective/surjective), but I think this version is still going to be hard going for those who aren't math students.

I agree that the previous example got unhelpfully big and wasn't the clearest. I was trying to explicitly show a set of ordered pairs. Your example seems fine, or we can still find a better one.

You are probably right that a bijection should be described primarily as a function, as this is the main way people think about it. I wonder though how we can make that version as clear as possible to laypeople.

The part about counting is helpful, but a bit awkwardly worded I think.

misconception that using common English words instead of the proper technical words may makes understanding easier – it doesn't make detailed technical understanding easier, but it certainly makes it easier for a newcomer who doesn't have any clue what the jargon words mean to figure out the basic idea. The new version has enough jargon in the first few paragraphs to be noticeably less welcoming.

The new paragraph about invertible functions helpfully includes an inline definition of inverse, but unfortunately it is not at all accessible to anyone who isn't a math student. It assumes people know not only about this arrow notation, but also about function composition and identity functions. I wonder if this paragraph can be made less obscure. –jacobolus (t) 16:15, 15 November 2023 (UTC)[reply]

This phrase seems to have either an extraneous word, or is missing a word: "that for each element of the codomain is mapped to from at least one element of the domain". DonFB (talk) 20:41, 15 November 2023 (UTC)[reply]

Fixed. D.Lazard (talk) 21:28, 15 November 2023 (UTC)[reply]

"to from"?

Somewhat related to the above comment, but "to from" in the first sentence initially looked like a typo to me: each element of the second set (the codomain) is mapped to from exactly one element of the first set (the domain)

After a few rereads, I worked out that it should (probably?) be parsed as "[is mapped to] [from]", but I wonder if it would be clearer to reverse the order: each element of the first set (the domain) maps to exactly one element of the second set (the codomain)

But I've been away from set theory for too long to be sure whether this is correct, so I'd appreciate a review from someone who better knows the subject. Achurch (talk) 06:38, 29 July 2024 (UTC)[reply]

Change done with a slight modification (it is the function that maps, not the element of the domain). D.Lazard (talk) 09:05, 29 July 2024 (UTC)[reply]

In the first paragraph, there are two statements defining bijection. The second statement starts with the word "equivalently". I believe that the second statement is in fact not equivalent to the first statement.

chownah 2403:6200:8853:1C5C:94F7:7747:A0FF:B048 (talk) 03:18, 1 September 2024 (UTC)[reply]

I agree: The first sentence "...is a function such that each element of the first set (the domain) is mapped to exactly one element of the second set (the codomain)." is the definition of a fucntion, while the second sentence "...a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set." is actually the definition of bijection. Hajijohn (talk) 15:17, 17 September 2024 (UTC)[reply]

This previously said: "A bijection [...] between two mathematical sets is a function such that each element of the second set (the codomain) is mapped to from exactly one element of the first set (the domain)." D.Lazard changed it recently, and I do not quite understand why. The new version seems incorrect. –jacobolus (t) 23:16, 17 September 2024 (UTC)[reply]

Ooops. The motivation of my change was to avoid the confusing (see above) "mapped to from". I restored the original version with "is mapped to from" replaced with "is the image of". D.Lazard (talk) 08:46, 18 September 2024 (UTC)[reply]

Color plot?

I created a few plots of the bijection function used here and added color and more detail than the current plot. The repository is public at my GitHub page.

Comments and edits welcomed, of course. soapbox (talk) 15:56, 10 February 2025 (UTC)[reply]

For being usable in Wikipedia, a plot must before be uploaded in Commons:, with the licensing information filled. D.Lazard (talk) 16:20, 10 February 2025 (UTC)[reply]

Apologies. It's located here. soapbox (talk) 22:18, 10 February 2025 (UTC)[reply]

I find this image pretty confusing. What are you trying to demonstrate with it? Why is this a line chart? –jacobolus (t) 23:38, 10 February 2025 (UTC)[reply]

The image is patterned after the existing plot of the same function on the page. I added color to make it visually easier to see the integer locations. Indeed it doesn't need to be a line chart.

File has been updated without interpolation lines. soapbox (talk) 20:34, 11 February 2025 (UTC)[reply]

"may be" is correctly used in the definition of bijection relations?

Hi.

In the section "Definition" of this article, the 2nd and 4th conditions of the definition of bijection relations with sets X and Y currently use "may be" instead of "must" like the 1st and 3rd conditions. What if to use "must" instead of "may be"? This replacement may clarify the definition.

"

each element of X must be paired with at least one element of Y,
no element of X may be paired with more than one element of Y,
each element of Y must be paired with at least one element of X, and
no element of Y may be paired with more than one element of X.

" Goodphy (talk) 07:20, 21 December 2025 (UTC)[reply]

This is basic English grammar; “no element must” does not work. ~2025-31850-11 (talk) 12:45, 27 December 2025 (UTC)[reply]