Question

Here is my first question, I'm preparing for the Analysis 1 exam and I found something that bothers me (not the first that shakes me I have to admit). If this is not a suitable question for mathoverflow just let me now, but I think that this can be of some help to people like me who are starting learning math.
I'm studying functions, and here is my problem:

$X = Y = {\mathbb N}$, and $f(n) = 2n$. Then I have $f({\mathbb N}) = 2{\mathbb N}$. If $m$ is even $f^{-1}(m) = {m/2}$, If m is odd $f^{-1}(m) = \emptyset$

The book I'm studying on says f is not surjective, how this can be?
I can say that the graphic of that function is as follows:

My only assumption is that:

$f(x) = y$ is not the same as saying $f(n) = n$

So I cannot use the graphic to understand the problem, but why is that?
By the way I think I'm still too focused on graphical representation of things, is this bad in math?

P.S.: Sorry for bad Latex formatting, this was my real first usage.

Answer 1

Well, the question about whether graphics representations of things is helpful is kind of a mixed bag. It's not absolutely bad to draw a picture to help understand the problem - actually for some areas it could be an excellent idea depending on how you learn. However the problem in this case is that the picture is misleading. Your picture shows an image which is really defined on all the real numbers, but if I understand the question correctly - the function is on the counting numbers (natural numbers) (i.e. 1, 2, 3, 4, etc).

It seems to me that's where your problem stems from as well - you are looking it at like it's on all the real numbers, not just the counting ones.

If we recall the definition of surjective:

In mathematics, a function is said to be surjective or onto if its range is equal to its codomain. A function f: X → Y is surjective if and only if for every y in the codomain Y there is at least one x in the domain X such that f(x) = y. A surjective function is called a surjection.

(stolen from wikipedia so I don't have to type it myself).

Then X here is N and Y is N and there is no number n in N such that 2 * n = 1, thus so no value satisfies f(n) = 1 and the function is not surjective.

In regards to your last question - my opinion (I'm not a site admin or anything) is that this would qualify as a pretty good question for the site.

Answer 2

Thanks to the magic of tikz, pdfcrop, pdftoppm, and pnmtopng, here's the graph that you should have drawn.