Colon-to notation

In math­e­mat­ics, the no­ta­tion \(f : X \to Y\) (here, “colon-to no­ta­tion,” be­cause the ar­row \(\to\) is writ­ten “\to” in LaTeX) means that \(f\) is a func­tion with do­main \(X\) and codomain \(Y\). It can be read “$f$, a func­tion from \(X\) to \(Y\).”

This can be thought of as as­cribing a func­tion type to the value \(f\). The use of a colon to ex­press that a given value has a given type, as is done in type the­ory, is a gen­er­al­iza­tion of this no­ta­tion.


\(f : \mathbb{R} \to \mathbb{R}\) means that \(f\) is a func­tion from the real num­bers to the real num­bers, such as \(x \mapsto x^2\) (map­sto no­ta­tion).

\(f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) means that \(f\) is a func­tion from pairs of real num­bers to real num­bers. The \(\times\) here refers to the Carte­sian product of sets.


  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.