Talk:Surjective function

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Reversible and Inverse[edit]

if a surjective function is revrsible, how does the first picture there represent a reversible function. The C appears to be mapped to by two elements of the domain; wouldn't that make it no surjective? anton — Preceding unsigned comment added by (talk) 15:39, 16 August 2007 (UTC)Reply[reply]

It's not very clear in the article. "Reversible" simply means that if g(C) = 4, f(4) = C. It doesn't matter that g(C) can also equal 3. I'm going to try to put that into the article. --KSnortum (talk) 20:52, 25 November 2007 (UTC)Reply[reply]

There seems to be a contradiction in the part "surjection and epimorphism". First it says that there is equivalence surjection and epimorphism , then it says that it holds only for the category of sets!! — Preceding unsigned comment added by (talk) 13:13, 26 August 2013 (UTC)Reply[reply]

Example of a actual function that is surjective but not bijective with image[edit]

I think the images here are exceptionally good. I might add an example of an actual function that is surjective, but not bijective such as 

Just noticed that this is a cool example of the product of two odd functions being an even function :) Lfahlberg (talk) 06:27, 29 June 2013 (UTC)Reply[reply]

Since nobody objected I added an example (without an image) of a surjective, but not bijective function.Lfahlberg (talk) 15:36, 6 August 2013 (UTC)Reply[reply]

Arrow Symbols[edit]

I like this page, but would someone please add a table of the different arrow symbols that are commonly used for onto, one-one, etc? It would also be nice to note the LaTeX commands for the symbols. Jfgrcar (talk) 20:02, 2 June 2014 (UTC)Reply[reply]

I just inserted a template which links to a page describing some of the notation. This does not explain all of the symbols but it should help. — Anita5192 (talk) 01:18, 3 June 2014 (UTC)Reply[reply]


According to the wiki-definition of a function, the title of this page is wrong.

A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: 
every element of X is the first component of one and only one ordered pair in the subset.

This, however, is obviously not surjective. The most used term in numerous books would be surjective map or mapping, but not function. Should we change the title? — ManniFaltig (talk) 20:47, 22 June 2014 (UTC)Reply[reply]

Your point is not clear. Not all functions are surjective (unless you are a member of that small minority who don't believe in the concept of a codomain – Y in the above definition.) Bill Cherowitzo (talk) 04:55, 23 June 2014 (UTC)Reply[reply]

Clarification needed for the pictures[edit]

In my understanding surjectivity allows the existence of members of the domain that are not mapped into any element of the codomain. This is not visible in any of the pictures on this page and neither on,_injection_and_surjection csavlovs , Sep 16 2016 — Preceding unsigned comment added by (talk) 06:32, 16 September 2016 (UTC)Reply[reply]

Nope. If there were elements of the domain that were not mapped to anything in the codomain, then this would not be a function (it would be a proper partial function) and this article is about surjective functions. --Bill Cherowitzo (talk) 16:12, 16 September 2016 (UTC)Reply[reply]

Gallery section created[edit]

I have moved the awkwardly placed images to the gallery section. This severely benefits the readability of the article. A line is included in the lead to make this easily apparent/accessible to all readers. This article has suffered from an unfortunate lack of quality for such a highly visited/essential topic. IaSpiralSource (talk)


Okay, I know I just created a section, but c'mon: do we really need a redirect for the word *onto!?* That is absolutely absurd. SpiralSource (talk)


  • Greek: Επιρριπτική συνάρτηση

(SURJECTIVE = επιρριπτική, INJECTIVE = ερριπτική, BIJECTIVE = αμφιρριπτική) — Preceding unsigned comment added by 2A02:587:4115:1000:B854:2FD1:F18D:DC09 (talk) 22:10, 18 July 2019 (UTC)Reply[reply]

Sloppy Definition[edit]

Please write "iff" and not "if",

and referencing a couple of web-sites instead of a math text book is not a good idea Jan Burse (talk) 10:30, 30 July 2022 (UTC)Reply[reply]

being surjective is not a property of a function as it is defined in the article about functions[edit]

According to Function (mathematics):

a function is a binary relation that is univalent and total.

If this is our definition of a function, then "surjective" is not a property of a function and can only be true of a function together with a set. I don't know, maybe something like "(f,B) is surjective iff im(f)=B". However, one can make sense of an expression like "f:A -> B is surjective", because a set B is explicitly mentioned. Imo this should be made more explicit in the article. 2A02:168:FE38:0:FCB6:AAED:7EDF:9D9 (talk) 14:39, 20 October 2023 (UTC)Reply[reply]

Your quote is not the primary definition given in this article. Moreover, the definition of a binary relation involves two sets. In the case of "a binary relation that is univalent and total", the first set is the domain, the second set is the codomain. There is no need of a third set for defining "surjective". D.Lazard (talk) 14:57, 20 October 2023 (UTC)Reply[reply]
<code>Your quote is not the primary definition given in this article.</code>
It is the only rigorous one. And even if it is not, you call it, the "primary" one, it is still there, and this article should address potential inconsistencies with the notion of surjectiveness.
Moreover, the definition of a binary relation involves two sets. In the case of "a binary relation that is univalent and total", the first set is the domain, the second set is the codomain. There is no need of a third set for defining "surjective".
First, the definition is given quite informally. Second, it does not include "the sets X and Y" if you don't include them in the definition. There is even a short discussion about that in the article about binary relations:
In order to specify the choices of the sets X and Y, some authors define a binary relation or correspondence as an ordered triple (X, Y, G), where G is a subset of �� called the graph of the binary relation.
Or please tell for the set {(1,2),(2,3)} if it is surjective or why it is not a function? 2A02:168:FE38:0:FCB6:AAED:7EDF:9D9 (talk) 09:48, 21 October 2023 (UTC)Reply[reply]
The notion of a binary relationship requires two sets to be meaningful at all, because the Cartesian product requires it. The set {(1, 2), (2, 3)} cannot even be determined to be total, let alone being a function, without that additional data. Just like being surjective requires a codomain to be meaningful, being total needs a domain to be meaningful.--Jasper Deng (talk) 10:42, 21 October 2023 (UTC)Reply[reply]
(edit conflict) In your last example, the set {(1,2),(2,3)} is simply a set of pairs of integers. For having a relation, you must specify X and Y. If either you have not a relation or the relation is not total; in both case you have not a function. If and you have a function which is surjective, and even bijective. If and you have a function that is not surjective.
By the way, you are right by saying that the only way to define functions formally is to define them as a relation. However, in mathematics, the formal definition is rarely sufficient to understand a concept, especially for "rich" concepts such as the one of a function. If you do not think of a function as a proccess that associates an element of the codomain to each element of the domain, you will never be able to use functions. D.Lazard (talk) 11:15, 21 October 2023 (UTC)Reply[reply]
I wanted to note, the definition of surjective function correspnding to 2A02's function definition is in Surjective function#Surjections as binary relations: "A surjective function with domain X and codomain Y is a binary relation between X and Y that is right-unique and both left-total and right-total." Mathnerd314159 (talk) 01:18, 22 October 2023 (UTC)Reply[reply]