# How to denote domain and range

Is there a common way to denote the domain or range of a function? I've seen things like $\cal{D}\left(f\right)$ and $\cal{R}\left(f\right)$ for the domain and range, respectively, but I wasn't sure if there was a more explicit syntax for this (e.g., the way that cardinality has a specific syntax instead of just $\mathrm{card}\left(X\right)$, for instance).

I know that you alternatively mention the domain and range in the "mapping" notation of the function, e.g.:

$$f: \cal{D} \to \cal{R}$$

But I'm really just looking for a way to explicitly indicate the domain/range of a given function.

Notations vary. You should pick something straightforward, and state it clearly up front. One common choice would be $$\operatorname{dom} f\\ \operatorname{ran} f$$

another reasonable choice might be $$\mathscr{D}(f)\\\mathscr{R}(f)$$

Principia Mathematica used:

Unfortunately the term range means different things for different people.

In a function $f: \cal{D} \to \cal{R}$, we call $\cal{R}$ the codomain of $f$; it's the set where $f$ takes its values.

The image of $f$ is the set of values of $f$; it's a subset of the codomain, but usually smaller.

The term range means either codomain or image, and so is better avoided.

If you need notation, you may use $\text{dom}(f)$, $\text{codom}(f)$, $\text{im}(f)$.

• I was about to type up the same response when yours popped up. This is certainly an important distinction. xamuel.com/range-vs-codomain gives some of the popular reasons. Also important to note the use of preimage for domain, though those terms also have some conflicting use. Jun 11, 2014 at 16:29