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Contents 1 To do 2 Easy way to remember which is which 3 Merging injective, surjective and bijective 4 One-to-one 5 == 5.1 bold edits 5.2 Definitions

[edit] To do

• latex to html the examples
• Evaluate see also links:
  • injective module does not seem relevant

Of course this list is non-exhaustive -MarSch 13:52, 5 May 2005 (UTC)

Besides the above,

• Mention about cases when a domain or a codomain is empty.

-- Taku July 9, 2005 14:02 (UTC)

[edit] Easy way to remember which is which

Maybe I'm just unfortunate in this regard, but everytime I read a paper that refers to a function being injective or surjective, I have to look up which is which. Does anyone have a mnemonic or some way to remember what each of these refer to? Even some explanation as to the origin of the names might be helpful.

-1nject1ve=(1 to 1)

This is how I have memorised these words: if a function f:X->Y is injective, then the image of the domain X is a subset in the codomain Y but not necessarily equal to the whole codomain (or, more precisely, a function f:X->Y is injective iff the function f defines a bijection between the set X and a subset in Y); as the word "sur" means "on" in French, "surjective" means that the domain X is mapped onto the codomain Y, i.e. the whole codomain is covered by the image of the domain. As the prefix "bi" means "two", "bijective" means a function that has both properties: is both injective and surjective.

[edit] Merging injective, surjective and bijective

I think merging the three pages was a very bad idea. Of course there was a certain overlap between those articles but I do not see how discussing them on one single page provides any benefit. This should be obivous as the merge created a second parallel hierarchy in the article namely the subheadings injective, surjective and bijective under each major heading. Besides increasing the length of the article this is very confusing for the reader. MathMartin 17:08, 14 May 2005 (UTC)

This stuff is so related that I thought it appropriate to put it together. Of course there is a heading for each species separately, but, as in other articles, headings do not imply that stuff should be in separate articles. The way the picture contrasts the 4 (not 3) possibilities should also give you a clue. That you can change the species of a function simply by changing its domain or codomain provides another justification. Because of the overlap between the three concepts this article is not much longer than any of the three separate ones were, which is why I wanted to do this in the first place. -MarSch 11:33, 15 May 2005 (UTC)

If it's grouped togethether because it's all related, surely it would all be better off as subsections under [[Map (mathematics)|Map]. I have taken the liberty of merging it in. Old page here --138.38.32.84 20:49, 15 January 2006 (UTC)

I have reverted the merge of this page into Map per the discussion at Talk:Function (mathematics). If you want it merged, take it to AFD. Perel 03:10, 6 December 2006 (UTC)

[edit] One-to-one

I just came here by the search query "one-to-one correspondence", which I expected to mean bijective. Now, the article only mentions 'one-to-one' as meaning injective, while e.g. [1] tells me, 'one-to-one correspondence' means in fact an bijective function. Could the article possibly say something about the usage of these terms?

"1-1" means injective; "1-1 correspondence" means bijective. Revolver 05:11, 10 September 2005 (UTC)

I believe that it was exactly this source of confusion that prompted adoption of the new terminology. --noösfractal 05:25, 10 September 2005 (UTC)

Maybe. It's not "new", really. The terms are due to Bourbaki. Revolver 06:44, 10 September 2005 (UTC)

[edit] ==

this page is awfully good. thank you to those who made it!

[edit] bold edits

I have made some fairly bold edits to this page, in an attempt to make it both more readable as well as well-organised and consistent. In particular, the way the definitions were worded in common language seemed very misleading and confusing to me. I know, it's true, we know what it should mean, but someone else not knowing this may read it differently. Even informal definitions should be intuitively "rigorous" and not transpose quantifiers at will. I also included substantial new material. I hope that some of the new material doesn't seem overly technical or abstract. These properties are actually extremely important and for injective/surjective, are dual to each other. Revolver 06:49, 10 September 2005 (UTC)

[edit] Definitions

The word argument, which appeared in the definition of injective/surjective/bijective means a parameter, not an element of a domain. A function can have just one parameter, which takes values from a domain of many elements.