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I'm having trouble keeping the definitions straight, so to help me remember the definitions: is it correct to say that a function is:

  • Injective (one-to-one) iff it is not many-to-one.

  • Surjective iff its codomain and range are equal.

?

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"Many-to-one" often has other, specific meanings: for example, a function is called "two-to-one" if every point in the image has exactly two preimages; while such a function is necessarily not injective, not every non-injective function is like that (e.g., $f(x)=x^2$ is not one-to-one, but it is also not two-to-one, because there is a unique point mapping to $0$). So I would avoid "many-to-one". Better is "different points map to different points". –  Arturo Magidin Feb 14 '11 at 14:45

3 Answers 3

up vote 2 down vote accepted

The first statement is true because if the function $f$ injective every set of two different elements $a,b$ is sent to different images $f(a),f(b)$ and if two different elements $a,b$ are sent to the same image $f(a)=f(b)$ the function is not injective.

The second statement is also true. That is exactly the definition of surjective as the image is identical with the set that is mapped onto, so each element is hit at least once. Note that the term "image" is better than range because range can also stand for codomain.

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Cool, thanks! :) –  Mehrdad Feb 14 '11 at 10:10

First of all, I cannot understand why it is tagged linear algebra instead of set-theory.
Then, when I was learning set theory, I had the same problem as @Mehrdad has, and in the end I found the most intuitive and useful definition is that of Bourbaki:
A function is injective if it has a retraction;
A function is surjective if it has a section,
whose definition is in here.
This definition is easy to imagine and useful in application both, hope you will be satisfied. As for the two definitions mentioned, they are as @user3123 stated.

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Hm... the problem with those definitions is that I don't know what retraction and section mean. :( Oh, and I tagged it linear-algebra since I came across this in my linear algebra book (about vector spaces and transformations.) But thanks nonetheless. :) –  Mehrdad Feb 14 '11 at 10:17
    
Well, the link goes to a page explaining in detail what retractions and sections are, besides, the set-theory by Bourbaki contains a great deal of discussions on them which might be helpful to you. Although you came across it in your linear-algebra book, you still can tag it as set-theory as well, thanks. –  awllower Feb 14 '11 at 13:36

As I mentioned in a comment, I would suggest shying away from "many-to-one." This often has a very specific interpretation which implies non-injectivity, but is not equivalent to non-injectivity. For instance, one says a function is "$n$-to-1" to mean that for every point $a$ in the image, there exist exactly $n$ distinct points $x_1,\ldots,x_n$ such that $f(x_1)=\cdots=f(x_n)=a$ (hence "1-to-1" means each element in the image has exactly one pre-image, i.e., injectivity). But this means that "many-to-one" is often used to mean that every point in the image has multiple pre-images.

Of course, any (nonempty) function that satisfies the condition that every point in the image has multiple pre-images is not injective, but the converse does not hold: the function $f\colon\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^2$ is not injective (since $f(1)=f(-1)$ and $1\neq -1$), but there is one and only one point of the domain that maps to $0$.

I would instead suggest

  • Injective (aka one-to-one) if and only if it sends different elements to different elements;
  • Surjective (aka onto) if and only if every element of the codomain is the image of someone. (Alternatively, as suggested by user3123, if and only if the image equals the codomain/range).

As far as remembering that "injective" means "one-to-one", you can think of a one-to-one function as a function that injects the domain into the codomain, so that you end up with a "copy" of the domain inside the codomain (much like an injection puts everything in the syringe into your body). "Surjective" comes from the French sur, meaning "onto"; or you can think of the Latin super (above, over), imagining that the domain is complete above and covering the codomain.

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The French was how I remembered surjection when I was just starting. –  BBischof Feb 15 '11 at 1:08
1  
@BBischof: I used a rather silly phrase the professor teaching the class suggested: "Los monos se inyectan con la izquierda" (monkeys inject themselves with their left hand, in Spanish); so monomorphisms are injective and can be cancelled on the left (the "mono" part was how to remember it was one-to-one). Surjections were just "the dual kind". (-: –  Arturo Magidin Feb 15 '11 at 4:00
    
that is pretty amusing, I STILL forget which side is cancellable sometimes, and I have to chck for an example to remember. Another somewhat embarrasing truth is that I forget direction of the arrows in prod/coprod one in a while. Again I work out a simple example to remember. –  BBischof Feb 15 '11 at 4:20

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