# 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.

# Examples

$$f : \mathbb{R} \to \mathbb{R}$$ 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.

Parents:

• Mathematics

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

• This no­ta­tion is ba­si­cally uni­ver­sal in math­e­mat­ics but I don’t know a name for it, so I made one up. I don’t have strong opinions about whether it makes sense for this to be its own page or whether there should be some big page of com­mon math­e­mat­i­cal no­ta­tion.