Talk:Bijection, injection and surjection

message: I have replaced all the images with available SVG equivalents. Superm401 - Talk 22:40, 26 October 2007 (UTC)
thread_title:  SVG 

message: I fixed a couple of minor formatting issues in this section, but I have other concerns.
On the first bullet, A = f −1(f(A)) and f(f −1(B)) = B seem ill-defined (with the exception of the injection).  This is because f is not invertible if it is not an bijection (though it can be reversible if it is an injection, which becomes an inversion if you simply restrict the domain of f −1 to the image of f), but in general f −1 is not well-defined without additional qualifications.  There's some unconscious type-raising going on here, in other words.
On the second bullet, I'm not familiar with the function descriptions of the format H : A → h(A) : a → h(a) (though it certainly looks as though somebody's laughing at me).  Perhaps this should more closely model the description of functional decomposition given in the injective function article?
Also, depending on how these are described, it may be necessary to mention that certain aspects (such as, the existence of an injection from A to B implies a surjection exists from B to A) are not always true if AC is not assumed.  I don't think the article on injective functions is direct enough in describing this, and I may mod it slightly.  TricksterWolf (talk) 20:51, 25 August 2011 (UTC)
replies: text: If A is a subset of the domain of function f, then f −1(A) denotes the preimage of A under f (which itself is a (maybe empty) set), so it is well defined. DerVanda (talk) 13:48, 17 March 2024 (UTC)
thread_title:  Minor concerns with "Properties" Section 

message: The diagram in the lead has one variable in italics and the others in regular typeface.  It would nice if someone could fix the one inconsistent image, Bijection.svg. — Anita5192 (talk) 19:58, 14 April 2016 (UTC)
replies: text: Fixed. George Makepeace (talk) 21:13, 14 April 2016 (UTC)
thread_title:  Lead diagram 

message: The two logical definitions of bijection, & surjection have three free variables. While they are in a common abbreviated form, they could be improved by quantifying the domain and co-domains, since order of substitution matters here.
I suggest an existential quantification at the start, which clarifies the definitions only need to hold for particular domains, rather than all. The talk page actually contains a question from 2011, about whether or not the co-domains are arbitrary. This suggests it's a commonly needed clarification.
Whole Oats (talk) 00:09, 29 January 2018 (UTC)
thread_title:  Free variables in the definition of Bijection and Surjection 

message: How do Wikipedians expect the reader to guess what the crap does it denote? And why don’t simply say  for algebraic examples and  for logarithm? Incnis Mrsi (talk) 11:50, 23 December 2018 (UTC)
thread_title:  '''R'''<sup>+</sup> 

message: I really thought the large diagrams in the centre of the page was very helpful for visualizing what each function means. If anyone cares I would vote to return the beginning of the page back to how it was instead of keeping the diagrams small and to the side.  — Preceding unsigned comment added by Koanarec (talk • contribs) 11:51, 9 April 2020 (UTC)
thread_title:  large pictures of each function 

message: I have concerns about the definition of injection. The wordy definition is not equivalent with the logical definition. Injection is defined as so: 
"The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain..."
The logical definition is
\forall x,x'\in X, x\neq x' \implies f(x) \neq f(x')  
Which states that for every element x and x' in codomain X the implication is true.
But the implication is true even when x and x' are equivalent but their images f(x) and f(x') are not. This can be seen from the implications identical disjunction representation  \neg (x\neq x') \lor f(x)\neq f(x')  which states that a function f is injection when  x=x'  or  f(x)\neq f(x')  is true.
This means that the logical definition of injection is saying that in injective mapping some element in codomain is mapped to more than one element in the domain, which contradicts the wordy definition.
I think this is a major issue, and should be resolved.
I suggest that the implication from the logical definition changes to double arrow  \iff . This resolves the issue and is in agreement with the wordy definition. So the new definition should be:
 \forall x,x'\in X, x\neq x' \iff f(x) \neq f(x') 
But because this is about the very core of mathematics, there should be a consensus before changing the definition.
Otacon.w. (talk) 08:33, 19 November 2020 (UTC)
replies: text: The case (x=x')\land (f(x)\ne f(x')) cannot occur because of the definition of  a function. So, the "wordy definition" is equivalent with the logical formula, and there is no reason for changing the formulation of the article. D.Lazard (talk) 09:20, 19 November 2020 (UTC)
thread_title:  Concerns about the definition of injection 

message: I question the validity of this section as written, because, as yet, we have not established that f exists.—Anita5192 (talk) 15:20, 17 March 2024 (UTC)
replies: text: An inverse function, denoted f^{-1}, exists only if f is bijective. But the inverse image of a subset Y of the codomain is also denoted f^{-1}(Y), and it is defined as f^{-1}(Y)=\{x\mid f(x)\in Y\}. So, the section is valid, but misplaced, since its natural place is undoubtedly in Image (mathematics): The second item is about the factorization of a function through its image. The first item is about the relations between images and inverse images. If moved there, it would deserve to be expanded to say that one has a Galois correspondence, since images and preserve inclusions and satisfy f\circ f^{-1}\circ f=f and f^{-1}\circ f\circ f^{-1}=f^{-1} applied to subsets of the domain and the codomain. D.Lazard (talk) 17:05, 17 March 2024 (UTC)
thread_title:  Properties