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Motivation: Today I first wondered about and later remembered why the set of all functions from a set $X$ to $Y$ is denoted $Y^X$. They wikipedia page gives the explaination

"The latter notation is motivated by the fact that, when $X$ and $Y$ are finite and of size $|X|$ and $|Y|$, then the number of functions $X \rightarrow Y$ is $|Y^X| = |Y|^{|X|}$. This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities."

I then remembered a series of introductory group theory results in which the notation suggests arithmetic division of some group theoretic quantities.

There are also some other practical notations, which do not try to resemble somewhat easier mathematical operations, but which are actually suggestive on a physical level. The arrow "$\rightarrow$" is an obvious example, but also the braket notation for dual vectors $\langle\phi|,|\psi\rangle$ in quantum mechanics, which complete each other when you build a scalar product. A related personal anecdote is that at one point I wondered about the arbitrariness of the arabic numeral symbols and I came up with symbols from which you could read off their quantitative value, their joint behaviour under "$+,-,*$" and "$:$" as well as their prime factors.

My question: What are other examples for advertently or suggestive notations, which are practical in the sense of the examples given above?

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The Leibniz notation for derivatives $\frac{d}{dx}$ is a good example. Relevant post: 21199. –  Srivatsan Dec 24 '11 at 20:12
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You might find examples here. Certainly "good" often means "suggestive" with regard to notation. –  Dylan Moreland Dec 24 '11 at 20:15
    
@Srivatsan: Very true, that was an obvious one. And Dylan Moreland: Cool, thank you for the link. There are good answers in that link, although most of the suggested notations there are abbreviations in the right position, not "natural" notations in the sense of my question. –  NikolajK Dec 24 '11 at 20:27
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@Nikolaj I agree. I just thought you might be interested in the thread. –  Dylan Moreland Dec 24 '11 at 20:30

3 Answers 3

Another example from combinatorics (which boils down to the example you cited): the power set of $S$ is often denoted $2^S$. Indeed, a subset of $S$ is just a function $S \to \{1,0\}=2$.

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Gauss's congruence notation $x \equiv y \bmod m$ opened up a whole new point of view for manipulations and reasoning in number theory.

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The integral sign is just an elongated $S$, standing for sum, suggesting its origin as the limit of finite sums.

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Here's a nice reference for this –  t.b. Dec 24 '11 at 21:30

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