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I know $\overset{!}{=}$ means "must be equal to." Would it be reasonable to conclude that $\overset{!}{\neq}$ means "must be equal to anything but", i.e. "cannot equal to"?



References:
  1. Notation for “should be equal to”
  2. Must be equal and other stacked math symbols
  3. What does ! above = mean [closed]
GPWR
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    I have never seen the notation you refer to . Please provide a reference. – lulu Mar 14 '22 at 20:42
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    I have never seen this notation before today, but it does appear here: https://math.stackexchange.com/questions/3597085 – 311411 Mar 14 '22 at 20:42
  • I have mostly just seen $=$ used for this, but that symbol is overloaded, so introducing $\stackrel{!}{=}$ isn't entirely a bad idea. Some people use $:=$ for definitions, this seems like the same idea. – Arthur Mar 14 '22 at 20:57
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    I use $\stackrel{!}{=}$ to emphasise surprise or enthusiasm when it would otherwise cause confusion to use $!$ on the RHS. The same goes for $\stackrel{!}{\neq}$ or even $\stackrel{!}{\le}$. What I'm getting at, I suppose, is that the notation doesn't have to be mathematical in nature. – Shaun Mar 14 '22 at 21:21
  • @lulu Done. Thank you for pointing that out. – GPWR Mar 15 '22 at 18:38
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    None of those citations really back up the meaning you wish. I do not believe this is a widely recognized notation for what you want. I don't even think it's mathematical notation...maybe pedagogical. As a couple of your references suggest, if I saw it I'd be inclined to read it as "Though you really don't expect this, the left hand is equal to the right!" – lulu Mar 15 '22 at 18:42
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    @Shaun: I've also used "$\stackrel{!}{=}$" for surprise or enthusiasm. (For instance, I might start a trig identity proof with "$\stackrel{?}{=}$", and end with "$\stackrel{!}{=}$".) Annoyingly, many programming languages take "!=" to mean "is not equal to", turning the symbol for emphasis into one of negation. Moreover, some of those same languages are such that particular a variable x might be "nil" vs storing an actual value, and have introduced the notation x! as an assurance of an actual value ... effectively restoring the emphatic nature of "!" (ie, "x is really there!"). Fun! – Blue Mar 15 '22 at 19:38

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