Talk:Bijection

message: I've heard that the set of rational numbers is supposed to be in one to one correspondence with the set of integers. How is this supposed to be? Isn't there an infinite number of rational numbers for each integer. (e.g. 1, ... 1.001, ... 1.1, ... 1.3145, ... etc. 2, ... 2.1, ... 2.2, ... 2.3, ... etc.)
:The countable set page outlines a scheme for creating a 1-1 correspondence between ordered pairs of non-negative integers and the set of natural numbers, N. Essentially, this maps the ordered pair (m,n) as follows:
::(m,n) \rightarrow \frac{m^2+2mn+n^2+3m+n+2}{2}
:... which gives:
::(0,0) \rightarrow 1
::(0,1) \rightarrow 2
::(1,0) \rightarrow 3
::(0,2) \rightarrow 4
::(1,1) \rightarrow 5
::(2,0) \rightarrow 6
::etc.
:The scheme can be extended to map ordered pairs of integers (positive or negative) to N. You can also map the set of rational numbers Q to a subset of the set of ordered pairs of integers - the rational m/n maps to the ordered pair (m,n) where m and n are co-prime. Putting these two maps toghether gives you a 1-1 correspondence between Q  and a sub-set of N. Gandalf61 09:00, 18 April 2006 (UTC)
replies: text: Wow, it makes absolutely no sense, but it works. Weeeeird. Linguofreak 18:12, 20 April 2006 (UTC)
replies: text: Right... the union of countably many countable sets is countable.  For a decimal representation to be a rational number, it must either terminate or repeat, and it can be shown that for each integer, there are only countably many ways to write decimals after it that either terminate or repeat.  So take the union of all those and it's countable.  You may have to be careful about repeating 9's, but they don't really matter anyways.  —Preceding unsigned comment added by 69.181.216.178 (talk) 23:05, 8 May 2009 (UTC)
thread_title:  Huh? 

message: 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)
replies: text: 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)
thread_title:  total function 

message: 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)
replies: text: 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)
replies: text:  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)
replies: text:  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)
thread_title:  Too technical 

message: 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)
replies: text: 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)
thread_title:  Baseball, anyone? 

message: There appears to be some disagreement about the caption on the composition diagram. Paul August is perfectly correct, the composition is a bijection. The anon editor may be thinking of this: if (g∘f) is a bijection, then (g∘f)-1 is also a bijection and (g∘f)-1 = f-1∘g-1, which would be a contradiction since g is not an injection, so its inverse is not even a function. The subtle fallacy here is that while (g∘f)-1 = f-1∘g-1 holds for arbitrary relations, it only holds for functions if both f and g are bijections. The situation depicted in the diagram actually shows why this is so. Is this an important enough point to bring up in the article? If it is, I can use some help in finding a citation that discusses the failure of the formula when one or the other of the functions is not a bijection. (I have a number of sources for the proof of the formula, but none of them discusses the necessity of the conditions.) Bill Cherowitzo  (talk) 04:24, 30 July 2013 (UTC)
thread_title:  Skirmish with diagram caption 

message: 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)
thread_title:  image 

message: 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)
thread_title: good job

message: 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)
replies: text: . You are correct; thanks. D.Lazard (talk) 16:17, 31 October 2023 (UTC)
thread_title:  Is the second sentence of this article correct?