8
$\begingroup$

$f \circ g$ is usually interpreted as $f(g(x))$ although, as Google shows, $g(f(x))$ is used frequently too. My question: Does anybody know who was the first mathematician to use this symbol and what was his interpretation?

  • 3
    $f\circ g$ is *always* interpreted as $f(g(x))$; more precisely, the rule for the function $f\circ g$ is $(f\circ g)(x)=f(g(x))$. $g(f(x))$ would correspond to $g\circ f$: $(g\circ f)(x)=g(f(x))$. $f\circ g$ and $g\circ f$ are generally different functions.2012-04-17
  • 8
    $f\circ g$ is _not_ interpreted as $f(g(x))$. Rather $(f\circ g)(x)$ is interpreted as $f(g(x))$, and $(f\circ g)(w)$ is interpreted as $f(g(w))$, and $(f\circ g)(5.2)$ is interpreted as $f(g(5.2))$, and so on.2012-04-17
  • 0
    [Wikipedia](http://en.wikipedia.org/wiki/Function_composition#Alternative_notations) mentions this but has no references.2012-04-17
  • 2
    Google shows 60000 results for "(f o g)(x)= g(f(x))" If you interpret from left to right, then f should be applied first. But this interpretation is obviously of a minority. Therefore I would be interested in the opinion of the inventor.2012-04-17
  • 7
    it depends on whether you write $f$ as $f(x)$ or $(x)f$. although the latter notation is rare, it does occur (often in the context of permutation mappings, or other algebraic contexts in which "we do as we parse").2012-04-17
  • 4
    If you put the symbol for a function after arguments, then f∘g means (x)(f∘g) or equivalently ((x)f)g which means the same thing as g(f(x)) when you put the symbol for a function before arguments.2012-04-17
  • 1
    @BelsaZarkin: if you look at the results of your search, you'll be hard pressed to find mathematics papers. The only academic things seems to be on computer science and engineering.2012-04-17
  • 1
    As an aside: $xf$ instead of $f(x)$ is useful when diagram chasing.2012-04-17
  • 3
    Back in graduate school, in our Galois Theory text, the author wrote $x^\sigma$ for the result of applying automorphism $\sigma$ to the element $x$. So naturally composition was $x^{\sigma\circ\tau} = (x^\sigma)^\tau$. That is: $\sigma\circ\tau$ means: first do $\sigma$, then do $\tau$.2012-04-17
  • 0
    @GEdgar: That seems to prevail in algebraic texts. Some authors write: let s be a mapping from M into N and t be a mapping from N into P, then the product st of s and t is defined by x^st = (x^s)^t from M into P. So first s and then t is applied.2012-04-17
  • 0
    The order of composition was discussed [at MO](http://mathoverflow.net/questions/18593/what-are-the-worst-notations-in-your-opinion/18607#18607), too.2012-04-21
  • 0
    @Martin: Thanks again. I read that now. (They could give a hint when closing a question.) But the information about who used "o" first and with which meaning has not yet been answered.2012-04-21

4 Answers 4