I've been doing an online course in discrete mathematics, and the notation $f\colon A\to B$ has come up a few times, and it has not been explained what it means. I tried searching for it on Google, but unfortunately you can't search with characters like $\to$. If anyone could clarify what it means, it would be greatly appreciated!
Edit: There have been a few answers so far about mapping – I'm sorry, but I'm a complete layman when it comes to this. Is there anyway someone could explain it a bit more?
Best Answer
$f:A\to B$ means $f$ is a function from $A$ to $B$.
Example:
$\begin{align*}f:\Bbb R& \to \Bbb R_+\\ x & \mapsto x^2\end{align*}$
You've certainly already seen functions defined as $f(x)=x^2$ but as you start doing more complicated things with functions, you need the "formula" plus two other things: the domain $A$ and the codomain $B$.
The reason fo that is that the function I defined above is not bijective (if I give you $f(x)$, you can not find $x$ because it could be $\sqrt{f(x)}$ or $-\sqrt{f(x)}$) but I can define another function that is bijective:
$\begin{align}g:\Bbb R_+& \to\Bbb R_+\\ x & \mapsto x^2\end{align}$
Because now you know the $x$ I took to form $f(x)$ is in $\Bbb R_+$ so it can not be $-\sqrt{f(x)}$ so it has to be $\sqrt{f(x)}$.
Therefore you can define
$h:\begin{array}{ll}\Bbb R_+& \to& \Bbb R_+\\ x & \mapsto & \sqrt{x}\end{array}$
And $h$ will be the inverse function of $g$ which we write as $g^{-1}=h$. Also note that $f$ does not have an inverse function.