Talk:Bijection

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[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] 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 ;) —The preceding unsigned comment was added by 80.238.227.222 (talk • contribs) 12:04, July 14, 2006 (UTC)

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

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