# 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.

• 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! Jul 27, 2012 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.) Jul 27, 2012 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). Jul 27, 2012 at 19:32
• I see a rotated form of $\displaystyle\; \; \stackrel{..}{\stackrel{!}{=}} \; \;$ pretty often... Jul 27, 2012 at 20:10
• I've both seen and used $\stackrel{!}{=}$ to denote a requirement. Jul 27, 2012 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...

• 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. Aug 15, 2012 at 12:47
• @Matt But isn't that more to define something rather than demand that equality holds? Aug 15, 2012 at 18:25
• Yes! Sorry, I misread the question. Aug 15, 2012 at 18:48