Talk:Bijection

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Start Class High Priority  Field: Foundations, logic, and set theory
One of the 500 most frequently viewed mathematics articles.

Contents

[edit] merge

I think we should merge this page with injective function and surjection. They are already using the same pictures. Any objections? What should be the name of the new page? MarSch 16:36, 21 Apr 2005 (UTC)

That's a tough one. Maybe properties of functions? Deco 04:44, 22 Apr 2005 (UTC)
Yes, I was thinking maybe it could simply be jection or injection, surjection and bijection or

in-, sur- and bijection or maybe injective, surjective and bijective or then again in-, sur- and bijective. Jection or maybe jective would make linking easy: injective, although it doesn't highlight as I expected/hoped. Maybe I should file that as a bug. -MarSch 12:42, 22 Apr 2005 (UTC)

Let's not make article titles that are meaningless on their own. The linking feature as it is now is useful for links like noncommutative, although I can't say I've seen such a link recently. Deco 16:42, 22 Apr 2005 (UTC)
By merging, interwiki linking has been made almost impossible, since most other wikipedias have separate articles. What is wrong with navigation boxes to link related articles? To prevent erroneous automated interwiki linking, I have changed the redirect to the Bijection-section of the page. This will have no effect on en:wikipedia, but will stop people on dozens of other wikipedias to correct automated interwiki links manually. -- Quistnix 07:25, 28 December 2005 (UTC)

[edit] Line 5

Check line 5, function succ??? I dont get it --'''Rohit''' 08:11, 20 January 2006 (UTC)


succ=successive. as in successive interger after n (n+1)


[edit] Self-contradiction in one-to-one correspondence (About the incomplete totality of the set of all prime natural numbers)

Essay moved to User:BenCawaling/Essay. Gandalf61 08:31, 14 April 2006 (UTC)

[edit] Huh?

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)
Wow, it makes absolutely no sense, but it works. Weeeeird. Linguofreak 18:12, 20 April 2006 (UTC)

[edit] Composition of function

Is this right? "A relation f from X to Y is a bijective function if and only if there exists another relation g from Y to X such that g o f is the identity function on X, and f o g is the identity function on Y." f:X to Y; g:Y to X. So, g o f is identity function on Y and f o g is identity function on X.Its written in the otherway round. Kindly verify.

Hello, g o f means "first do f, then do g". This is the usual convention in mathematics. So f takes you from X to Y, and g takes you from Y back to X. So g o f goes from X to X. Hope that clarifies things. Sam Staton (talk) 09:53, 5 March 2008 (UTC)

[edit] 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)

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

[edit] permutation

This page says that bijection is also called permutation, while the permutation page says that permutation has to be on finite domain. Mizar does not restrict permutations to finite domains (http://mmlquery.mizar.org/mml/4.66.942/funct_2.html#NM2), but I don't claim that the terminology is right there. JosefUrban 23:14, 24 November 2006 (UTC)

[edit] Bijection Composition

I have moved this content from the top of this page:

In this page it is said that When X and Y are both the real line R, then a bijective function f: R → R can be visualized as one whose graph is intersected exactly once by any horizontal line.

For intuitively sound functions this is probably true.

However, consider the function f:R − > R with

f(x) = 2x for every x \in Q and

f(x) = − 2x for every x \in R-Q.

For this example it is ofcourse still true that for every y \in R there is one unique x with the property that f(x) = y. But it is not any longer possible to give a clear visualisation of this, in the way described in this page.


Regards, Bob v. R.

The theorem still holds for this example. The plot of this will look like a "X" going through the origin, but each arm of the x is not a continuous line, it is a "dense" set of of points. For all horizontal lines corresponding to rationals, the intersecting point will be on the line f(x) = 2x, and horizontal line line will not intersect the line f(x) = -2x, and a similar scenario occurs for horizontal lines at heights corresponding to numbers not in Q.--Nappyrash 09:28, 26 November 2006 (UTC)

[edit] Rename?

I might sound like a pedant (or perhaps even a moron!). But what are peoples opinions on renaming this page Bijective function? I only suggest this as we have pages Injective function and Surjective function, why not Bijective function? I only ask here, as I'm not too sure how to go about changing pages and redirects/I don't want to piss anybody off (especially people much more knowledgeable than me). Help plz 17:46, 14 January 2007 (UTC)

The reason there is an injective function term is because the term injection has multiple meanings. The terms bijection and surjection only have one (well-known) meaning. At any rate, bijective function redirects to bijection, so all the terms and their variants are covered. — Loadmaster 22:26, 1 March 2007 (UTC)
It's true that bijection is unambiguous, so there's no real need to have the article at bijective function, but seeing as injective function has to be at that title, wouldn't it make sense to have all three matching? Surjection already redirects to surjective function; why not make this one match the other two? Is there a good reason to be inconsistent? -GTBacchus(talk) 21:23, 9 April 2007 (UTC)
I have at least one argument against using "bijective function" as the main name: it's not at all common in actual mathematical writing! "Bijective map" is definitely more widespread (outside of pure nomenclature) and "bijection" wins over both of them. Similar usage patterns seem to hold for "surjective function"/"surjective map"/"surjection" and "injective function"/"injective map"/"injection". Arcfrk 03:57, 10 April 2007 (UTC)
Ok, that indicates moving the injection article to injective map, and our question is whether the others should accord with it, or use the more common bijection and surjection. I haven't got a strong opinion on that question; I guess I would lean towards consistency. -GTBacchus(talk) 06:04, 10 April 2007 (UTC)

[edit] Word origin

Speaking of names, does anyone know the etymology of the term "bijection" (and "surjection" and "injection")? I can't find any mention at dictionary.reference.com or at Merriam-Webster.com. — Loadmaster 22:26, 1 March 2007 (UTC)

Well, the terminology is due to Bourbaki, if that helps. And 'sur' is a Latin/French/English prefix for 'onto', so that's clear enough. And I've always assumed the 'bi' prefix alludes to the function being two-way i.e. invertible. No idea where they got 'in' or 'jection' from though. Algebraist 14:47, 16 March 2007 (UTC)
We have English "onto", telling us every element of a codomain is covered by some element of the domain, which corresponds to "surjection", and to "epimorphism". You'll never guess what "epi-" means in Greek. Then we have 1-1 with the meaning that distinct elements in the domain map to distinct elements in the codomain, which corresponds to "injection", and to "monomorphism". Of course, "mono-" is Greek for one. The reasoning behind "injection" is that the map essentially creates an identical copy of the domain inside the codomain; it "injects" it, with the "-ject" part deriving from the Latin jacere, to throw. Finally, we have one-to-one (which I always thought was horrible, too easily confused with "1-1"), which corresponds to "bijection", and to "isomorphism". We can understand the "bi-", meaning two, as either "injection"+"surjection", or (better) as indicating that the mapping is invertible, so we can map in two directions. We can translate "iso-" as "same", so isomorphic objects have the "same form". (The Greek μορφή means form or shape.) Of course, a "morphism" is a structure-preserving mapping, which I think is shortened from "homomorphism" (and "homeomorphism").
A good place to find etymology for a word is the American Heritage Dictionary, available at Bartleby or through Dictionary.com as well as OneLook (and other sites, I'm sure). A standard lexicon for Ancient Greek is LSJ (where we find that Morpheus, the god of dreams, gets his name from the shapes he conjures up in our heads).
The "-jection" names tend to be used for mappings of sets, whereas the "-morphism" names are used more in abstract algebra and higher mathematics. My impression is that the other names become less common as we move into university and beyond. --KSmrqT 08:15, 10 April 2007 (UTC)

[edit] vertical bar?

Excuse me, I am unable to understand the notation in:

  • For a subset A of the domain and subset B of the codomain we have:
|f(A)| = |A| and |f−1(B)| = |B|. —Preceding unsigned comment added by 83.54.90.204 (talk) 17:08, 2 April 2008 (UTC)