# 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.