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I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:

  • Iverson introduced the notation [X] to mean 1 if X is true and 0 otherwise; so for example Σ1≤n<x [n prime] is the number of primes less than x, and the unmemorable and confusing Kronecker delta function δn becomes [n=0]. (A similar convention is used in the C programming language.)

  • The function taking x to x sin(x) can be denoted by x ↦ x sin(x). This has the same meaning as the lambda calculus notation λx.x sin(x) but seems easier to understand and use, and is less confusing than the usual convention of just writing x sin(x), which is ambiguous: it could also stand for a number.

  • I find calculations with Homs and ⊗ easier to follow if I write Hom(A,B) as A→B. Similarly writing BA for the set of functions from A to B is really confusing, and I find it much easier to write this set as A→B.

  • Conway's notation for orbifolds almost trivializes the classification of wallpaper groups.

Has anyone come across any more similar examples of good notation that should be better known? (Excluding standard well known examples such as commutative diagrams, Hindu-Arabic numerals, etc.)

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28  
this question is broadly useful, so perhaps better as community wiki? –  Suvrit Oct 20 '10 at 20:09
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In set theory we write ${}^B A$ for the set of functions from $B$ to $A$. –  Andres Caicedo Oct 20 '10 at 20:54
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I've always assumed that the notation $A^B$ is because of the "exponential law" $(A^B)^C = A^{B\times C}$ ... –  Kevin H. Lin Oct 20 '10 at 23:18
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Arabic numerals ? Ah yes, they were transmitted to Europe by the Arabs. –  Chandan Singh Dalawat Oct 21 '10 at 3:29
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Isn't $x \mapsto f(x)$ commonplace? As for homomorphisms, they are not simply maps, and $\mathrm{Hom}(A, B)$ denotes the whole class, while $A \to B$ denotes a single mapping. –  Alexei Averchenko Oct 21 '10 at 3:38

64 Answers 64

I like the notation $A\mathrel{\in\in}\mathcal C$ for objects (rather than morphisms) in a category (not my own invention).

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Many years ago, I invented the notation $\mathcal{Lex}_{T}(X)$ as the function equal to $1$ if and only if the consideration of $X$ in the theory $T$ entails no contradiction and $0$ otherwise. Lex is both "law" in Latin and a shortcut for "Logical existence", which in some sense is the only law of mathematics.

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I like the notation $f:A\cong\subseteq B$ for "$f$ is an embedding of $A$ into $B$." The idea is that the relation of embeddability is obtained by composing the relations "isomorphic to" and "substructure of."

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@David: I tend to use $\hookrightarrow$ for maps that are literally inclusions. –  Andreas Blass Nov 25 '11 at 4:00
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I've mostly seen people use $\subset$ or $\subseteq$ for literal inclusions and $\hookrightarrow$ for embeddings (or whatever kind of injection is suitable). Reserving $\hookrightarrow$ for literal inclusions seems kind of pointless when $\subset$ exists. –  Ketil Tveiten Nov 25 '11 at 8:34
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... and then you can decorate the arrow with o or | to incorporate meaning like "open immersion" or "closed immersion". –  Konrad Voelkel Dec 19 '11 at 10:43

$\sin^{-1}(x)$ as opposed to $\text{arcsin}(x)$. This encapsulates the fact that it is an inverse function.

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16  
And, combined with $sin^2(x)$ for $(sin(x))^2$, it make one ponder the meaning of $sin^n(x)$ when $n=-1$. –  Harald Hanche-Olsen Oct 20 '10 at 20:10
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I have to go with this notation being anti-useful, since it is in fact not an inverse function except on a restricted domain (and that domain is different than the one for $\cos^{-1}$). Not that I have much call for it, but I decided once that I would never use $\sin^{-1}$. It takes greater fortitude to abandon $\sin^2$. –  Ryan Reich Oct 20 '10 at 20:17
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I have also permanently abandoned $\sin^{-1}$ after having numerous students mysteriously convert an $\arcsin$ into a $\csc$ without realizing it. (Of course, the notations $\sec$, $\csc$ and $\cot$ are almost completely worthless themselves.) –  JBL Oct 20 '10 at 21:34
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On this subject, $\arcsin(x)$ is actually great notation because it reminds you what the restricted domain is. That is, it gives outputs which are lengths of arcs (measured, as always, from the positive x-axis). Of course, this is never mentioned. –  Ryan Reich Oct 20 '10 at 21:56
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Gauss raged against the deplorable sin$^2x$ notation more than 150 years ago (sorry, I don't have the reference.) I guess we're stuck with it, though, since it's concise and ubiquitous in school maths. Also, how often do we really need the multiple composition of trig functions? –  John Bentin Oct 21 '10 at 11:37

protected by François G. Dorais Jul 9 '13 at 16:41

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