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Is there a formal notation to distinguish an equality that is a true statement, e.g.,

[...] and hence, $$ x = x^2 - 1 $$ [...]

from a demand, e.g.,

[...] so we require $$ x \stackrel{!}{=} x^2 - 1 $$ [...]

?

The same thing could apply to membership to sets $x\stackrel{!}{\in}\mathbb{R}$ and more.

I've seen the exclamation mark syntax once, and I faintly remember having seen some other notation, but I'm not sure if any of this is commonly used.

Nico Schlömer
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    I've once seen $\stackrel{?}{=}$ in Concrete Mathematics to represent an equality we are trying to prove/disprove; but I've not encountered notation like this elsewhere. +1 for the interesting question! – Thomas Russell Jul 27 '12 at 19:25
  • I have sometimes used this notation in my private notes. I suppose I may have learned it from somewhere, but if so I don't remember, and it sure isn't very common or standardized. (Though I won't rule out the existence of subcultures where it is common.) – Harald Hanche-Olsen Jul 27 '12 at 19:31
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    I've seen $\equiv$ used for identity (i.e. true for all values of the argument) as opposed to equation (which may or may not be true for any values of the argument). – gt6989b Jul 27 '12 at 19:32
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    I see a rotated form of $\displaystyle; ; \stackrel{..}{\stackrel{!}{=}} ; ;$ pretty often... – draks ... Jul 27 '12 at 20:10
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    I've both seen and used $\stackrel{!}{=}$ to denote a requirement. – joriki Jul 27 '12 at 20:53
  • I can imagine this being useful in a lecture, to help the audience follow the logic between formulas written on the board. I don't see much need for it in a paper: I'd rather have the meaning conveyed to me by the text surrounding the formulas. –  Jul 29 '12 at 00:52

1 Answers1

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Basically if I were you I would write the exclamation mark. If you fear that it is not understandable then remark in your text that this should specify that it is a demand and not a statement.

I think a lot of mathematicians use the exclamation mark in the sense you think of it. However, I don't think that it is "official" notation (like $e$ for the Euler number or so...).

(Although it doesn't exactly go for the question itself as the following deals with statements rather than demands I want to shortly mention here an)

Interesting note: Frege (the founder of modern logic) introduced a special sign to indicate that what follows it is an assertion rather than a truth. This sign is "$\vdash$".

So he would write at the beginning of a proof:

$\vdash 1+1=2$ in $\mathbb{R}$.

To say that he states that $1+1=2$ in $\mathbb{R}$ is true. Contrariwise

$1+1=2$ in $\mathbb{R}$,

is for Frege a truth value (or to be more accuarte: the Truth itself.)

Actually people in mathematical logic use this very sign to indicate tautologies. However, I don't really know if there is any connection between this use and Frege...

AndreasS
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