What are usual notations for surjective, injective and bijective functions?

I personnaly use $\hookrightarrow$ to mean injection and $\twoheadrightarrow$ to mean surjection. Although I do not have a particular notation to mean bijection, I use $\leftrightarrow$ to mean bijective correspondance.

My favorites are $\rightarrowtail$ for an injection and $\twoheadrightarrow$ for a surjection. In the days of typesetting, before LaTeX took over, you could combine these in an arrow with two heads and one tail for a bijection. Perhaps someone else knows the LaTeX for this.

I usually use two types of notations for function, injection, surjection and bijiection as follows.

Note that the \twoheadrightarrowtail is defined as follows, and the others are AMS symbols.

\usepackage{mathtools} \newcommand{\twoheadrightarrowtail}\mathrel{\mathrlap{\rightarrowtail}}\mathrel{\mkern2mu\twoheadrightarrow}}