Talk:Injective function

message: I don't like the title 'injections are invertible'.  To me invertible means to
have an inverse, (what the paragraph I am critisising calls a 'full inverse').
I would prefer something like 'injections have left inverses' or maybe 'injections
are left-invertible'.  Then the section
on bijections could have 'bijections are invertible', and the section on surjections
could have 'surjections have right inverses'. — Preceding unsigned comment added by 80.229.247.11 (talk) 06:02, 14 April 2006 (UTC)
replies: text: Perhaps in high-school, but the formal term injection is quite regular in even introductory college courses. --Jeff Wheeler (talk) 22:41, 15 December 2010 (UTC)
thread_title:  I don't like the title 

message: How frequent is the notation f: X ↣ Y ? I'm a professional mathematician and I never encountered it. On the other hand I've sometimes seen f:X\hookrightarrow Y. 82.229.188.151 (talk) 16:07, 12 March 2012 (UTC)
thread_title:  Notation 

message: Sorry, are the number of injective functions from X to Y (m in X, n in Y) n^m? I thought that was functions in general. What about (n!/((n-m)!)) ?
Physicproducer (talk) 00:26, 21 November 2012 (UTC)
thread_title:  Number of injective functions 

message: The 3rd paragraph has real problems, I think.  I'm not a mathematician, or even skilled in the art but please consider the following.
"A function f that is not injective is sometimes called many-to-one. However, this terminology is also sometimes used to mean "single-valued", i.e., each argument is mapped to at most one value; this is the case for any function, but is used to stress the opposition with multi-valued functions."
(as of Nov 1, 2013)
1. I find "this terminology" confusing, in the second sentence.  The entire first sentence is jargon laden so which "terminology" is being specified?  I do know what the sentence means, but believe it WILL confuse many others...Why not say something like: "However "many-to-one" is sometimes used to mean "all-to-one", i.e. all arguments are mapped to the same single value."  I also suggest removal of the "at most", since, in my ignorance, I don't believe you map can an argument to null, so it is NOT "at most one", it is "exactly one".  I am not confident enough in my knowledge here to make the change.  Someone help, please?
2.  The second part is really terrible and must be fixed.  It seems to have been written by a non-English speaker (non-native?).  Opposition???? This is awful.
First: WHAT is used to stress the opposition?  Second "stress the opposition" is virtually incomprehensible and terrible usage (did author mean "stress the contrast" or simply "contrast x with"? (I am not clear what is supposedly "used" here). Third use of the phrase "multivalued functions" is really going to confuse anyone that knows that functions are NOT multivalued. (see Multivalued_function).  It seems this clause is saying that even though it isn't saying anything, it is used anyway.
I am removing the clause, perhaps someone can fix what was being communicated and add another sentence here about contrast with multivalued functions (which are not functions).  Thanks.Abitslow (talk) 23:33, 1 November 2013 (UTC)
thread_title:  I think an edit is required, opinions? 

message: Injective relation redirects to this article. A function is a particular type of relation and the injective property is the same for both. Inclusive language could define injective for a relation, then apply the same condition for an injective function. The influence of software pushes mathematics toward relations, though the general theory has had a century and a half to develop. As textbooks such as those by Gunther Schmidt become known, the narrow focus on univalent relations (functions) will be broadened. — Rgdboer (talk) 23:38, 5 September 2018 (UTC)
thread_title: Relations

message: I didn't except anyone would dispute this point, so to prevent stubbornness or edit-warring, here are 38 sources (17 links) explaining that choice isn't needed for left inverses of nonempty injections, or to choose ONE element from a single set, or even one element from each set in a finite family of nonempty sets, even if those are infinite since it follows from a deduction rule of first-order logic:
1. Math.SE: "That is, there is no need for the Axiom of Choice in order to select an element form finitely many nonempty sets. In particular, you do not need the Axiom of Choice to show that you can choose a real number (a single set)."
2-6. MO: See answers 1-4 + Mummert's comment ("I think your first point is the main one for the question. It's alluding to the "existential elimination" inference rule, which says that from the assumption ∃xA(x), you can assert A(c) for some new constant c. The soundness of the rule is a direct consequence of the way semantics for first-order logic are defined."); 4: "The Axiom of Choice does not allow to "choose" a choice function, it only says that a choice function exists. To show that a choice function for a single nonempty set exists, you do not need to "choose" an element in the set, it is enough to show that at least one element exists (i.e. the set is nonempty). Every element in the set will give a different choice function."
7. Annoying Precision: "On the other hand, every monomorphism in Set (edit: with non-empty domain!) is split with no set-theoretic assumptions"
8. Choice function: "If X is a finite set of nonempty sets, then one can construct a choice function for X  by picking one element from each member of X. This requires only finitely many choices, so neither AC or ACω is needed."
9-10. MO: In addition to answer 1, see JDH's comment.
11-16. Math.SE: See answers 1-5 (and many of the comments): 1: "Knowing that S is non-empty, there do exist such x, and so we can declare the symbol x to denote an element of S, and the rest of our argument is then a function of x."; 2: "The ability to give a name to some arbitrary element of a nonempty set is known as "existential instantiation". This is an inference rule of the underlying logic, not part of the theory, and so it is included not only in ZFC, but also in ZF, and in appropriate forms it is also included in PA and every other first-order theory."; 4: "The axiom of choice, as said, allows us to choose from infinitely many sets at once (sometimes even when these sets are finite!). If, however, one only wishes to choose from finitely many sets then this can be done without it. It does not even matter if the sets themselves are finite or not."
17-22. Math.SE: See answers 1-6. 1: "It's a matter of the basic rules of inference allowed in proofs." 3: "the main answer is that making a single choice out of a single bin is a matter of logic. Specialized to this particular situation, if S is a set and you have proven ∃x:x∈S, then logic allows you to introduce a new constant symbol (say, a) along with the corresponding an axiom a∈S."
23. Quora: "The axiom of choice is still required to prove that a choice function exists for an infinite family of finite sets. That a choice function exists for a finite family of sets (of whatever size) can be proven in ZF without the axiom of choice."
24. Math.SE: First answer.
25-28. Math.SE: See answers 1-4.
29. Math.SE: See first answer: "No, you're not. The axiom of choice isn't relevant for making one single choice. The axiom of choice only becomes relevant when you have to make infinitely many choices."
30-32. Math.SE: See first answer + comments of Bananach and Asaf ("Choosing one element from a provably non-empty set has nothing to do with the axiom of choice, and everything to do with the inference rules of your logic.")
33. Math.SE: First answer.
34-35. Math.SE: See answers 1-2. "As you can see, no AC used here, because (morally) you don't need it to pick exactly one element from a set."
36. Quora: "just map it to some fixed, arbitrarily chosen element of A. There seems to be “choice” involved here, but in fact it is trivial: you need to pick just a single element of a single given set."
37. Math.SE: 1st answer: "One direction ... is just true without choice. If there is an injection f:X→Y, then there is always a surjection from Y onto X, … (here we used the fact that X is non-empty)."
38. Axiom of choice, at least three places in the page text: "In many cases, such a selection can be made without invoking the axiom of choice; this is in particular the case if the number of sets is finite", "… the axiom of choice … thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of the Zermelo–Fraenkel set theory without the axiom of choice (ZF); it is easily proved by mathematical induction.[6] In the even simpler case of a collection of one set, ... the axiom of choice says that every nonempty set has an element; this holds trivially.", "For finite sets X, the axiom of choice follows from the other axioms of set theory. In that case it is equivalent to saying that if we have several (a finite number of) boxes, each containing at least one item, then we can choose exactly one item from each box."
2601:42:0:4C76:7465:467D:C2F9:6CEE (talk) 07:12, 8 September 2018 (UTC)
thread_title:  38 sources for "every injection from a nonempty set has a left inverse without choice" 

message: If "univalent function" is essentially a synonym for "injective function" in complex analysis, should it be merged here? (Not to be confused with univalent relation, which is a kind of binary relation.) Apocheir (talk) 21:36, 31 August 2022 (UTC)
replies: text: The term Univalent function is used in three paper titles found in references for De Branges' theorem. See Talk:Univalent function for the "redundant but traditional" status of the term. As noted, univalent relations include functions, but the use in this complex analysis context actually refers to univalence of the converse of the function. Since the conflicting terminology is confined a narrow segment of mathematics, perhaps retaining the separation would prevent unnecessary spread of the conflicting terminology. See references Fitzgerald & Pommerenke (1985), Grinshpan (2002), and Milin (1977). Rgdboer (talk) 03:51, 3 September 2022 (UTC)
replies: text: Closing, given the uncontested objection with stale discussion. Klbrain (talk) 10:00, 20 November 2022 (UTC)
thread_title:  Proposed merge of [[Univalent function]] into [[Injective function]] 

message: Not sure if this is intentional, but the image displayed in the thumbnail when you Google injective map is not correct. It shows the example "A non-injective non-surjective function (also not a bijection)". Is it possible to change this? AcuteTriceratops (talk) 19:02, 15 October 2022 (UTC)
replies: text: I edited the article for having an injection as its first image. Hoping that this will fix your problem eventually. D.Lazard (talk) 08:25, 16 October 2022 (UTC)
thread_title:  Misleading Google Image result