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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.

Zev Chonoles
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jII
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  • I think f(x,y) is just a simplified notation for f( (x,y) ) and nothing more – Integral Feb 28 '13 at 00:22
  • And so, where should the line be drawn? What if $x$ in your line above was itself an ordered pair? – jII Feb 28 '13 at 00:26
  • there is no problem if that is the case, i could also write f(x) and say that x is a matrix, a polynomial or anything i want – Integral Feb 28 '13 at 00:31
  • "if we focus on the fact that a function is really a binary relation on two sets...." Don't focus on this fact. – Qiaochu Yuan Feb 28 '13 at 04:49
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    @QiaochuYuan, sometimes this fact is crucial to analyzing the situation in which such a function may arise. – jII Feb 28 '13 at 22:43
  • Notation varies widely, and it all really comes down to the same thing: you are specifying the name of the function and the point at which it is being evaluated. The details of notation are superfluous, as long as they are consistent it doesn't matter. For example, one often sees $Tx$ for what might better be denoted $T(x)$, especially where linear operators are concerned. And, believe it or not, even the notations $xT$ and $x_T$ and $T|_x$ are not uncommon, as are many other variants. – MPW Jul 21 '17 at 17:42

2 Answers2

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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.

Zev Chonoles
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  • ...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 (T v) written for the image of an element (v ∈ V) under (T).

    — Yes, this is using matrix T as, rather than an ordinary function, as a unary operator and using Prefix / Polish Notation (PN) — as opposed to Suffix / Postfix / Reverse Polish Notation (RPN)

     – cmarangu Jul 31 '23 at 01:13
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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.

Brian M. Scott
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