# Formal notation for a required statement

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.

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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! – Shaktal 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
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
I see a rotated form of $\displaystyle\; \; \stackrel{..}{\stackrel{!}{=}} \; \;$ pretty often... – draks ... Jul 27 '12 at 20:10
I've both seen and used $\stackrel{!}{=}$ to denote a requirement. – joriki Jul 27 '12 at 20:53

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...

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I like the way it's done in computer science: $$e := \lim_{n \to \infty} \left ( 1 + \frac{1}{n} \right )^n$$ where $:=$ denotes variable assignment. It doesn't look beautiful but it leaves the reader with no doubt. – Rudy the Reindeer Aug 15 '12 at 12:47
@Matt But isn't that more to define something rather than demand that equality holds? – AndreasS Aug 15 '12 at 18:25
Yes! Sorry, I misread the question. – Rudy the Reindeer Aug 15 '12 at 18:48