Is there a difference between writing $f: X\rightarrow Y$ and writing $f:X\mapsto Y$?

Question

I think I've heard about a year ago that "$\mapsto$" is only used for a bijection, or do they mean the same thing?

Answer 1

Most authors use $\rightarrow$ for a general function to denote that it is mapping elements from its specified domain to its specified codomain. Other, very similar notation is sometimes used for functions with special properties. For example, $\twoheadrightarrow$ is sometimes used to emphasize that the function is surjective, and $\rightarrowtail$ to emphasize injectivity.

On the other hand, if the function has been explicitly defined, $\mapsto$ is used to indicate where an element in the domain gets sent. For example:

If we have a function $f:\mathbb{R} \rightarrow \mathbb{R}$, mapping real numbers to real numbers, we can explicitly define it as follows: $$ x \mapsto x + 1$$

That is, $x$ is getting sent to $x + 1$.