What is the correct notation for a multivariable function?

Question

Many mathematical texts define a multivariable function $f$ in the following way

$$f := f(x,y)$$

However, if we focus on the fact that a function is really a binary relation on two sets, (say the real numbers), the definition would be as follows

$$ f: \mathbb{R}^2 \rightarrow \mathbb{R}$$

This seems to imply that the domain of the function is a set of ordered pairs of the form $(x,y)$.

The set $\mathrm{graph}(f) \subset \mathbb{R}^2 \times \mathbb{R}$, would then comprise ordered pairs of the form $$\left\{((x_0,y_0),a), ( (x_1,y_1),b),\ldots\right\}$$

In line with this, does it not follow that the correction notation for $f$ should be be

$$f := f( (x,y))$$

Few, if any, texts I have come across use this notation, although it appears the correct way to consider the function as a mapping from one set to another.

Answer 1

Technically speaking, you are correct. If I have a function $f:\mathbb{R}^2\to\mathbb{R}$, and I define $p\in\mathbb{R}^2$ to be the point $p=(1,2)$, then logically, there shouldn't be any difference between writing $f(p)$ or $f((1,2))$; and indeed, there is no logical difference. However, $f(1,2)$ is a convenient shorthand, because the extra pair of parentheses provide no added clarity.

I might also add that, in some places in mathematics, it is common to denote the application of a function without parentheses; for example, if $T:V\to W$ is a linear map from a vector space $V$ to a vector space $W$, you'll often see $Tv$ written for the image of an element $v\in V$ under $T$.

Choosing when to distinguish, or not distinguish, things that have "canonical" identifications, such as ordered pairs $((a,b),c)$ and ordered triples $(a,b,c)$, is an important detail in a lot of mathematics.

Answer 2

Yes, it’s formally correct. In most contexts, however, the added notational burden serves no useful purpose and merely makes the mathematics harder to read, which is contrary to one of the most important functions of good notation. In most calculus settings, for instance, it makes at least as much sense to think of the two arguments of $f:\Bbb R^2\to\Bbb R$ as independent entities $x$ and $y$, not as elements of a single compound entity $\langle x,y\rangle$. There are times, however, especially in set-theoretic contexts, in which it really is important to think of an input as a an ordered pair or $n$-tuple, and then it helps to reflect this in the notation.