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Consider a partial function $f$ that is defined only for a few values of its domain (my exact use case is $\delta$ transition functions in automata). One can 'complete' it by saying $$g(x)=0\iff f(x) \text{ is not defined.}$$

Is there a symbol to mean "undefined"? Would it be correct, or accurate, to write $\nexists f(x)$?

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Some people write $f(x)\uparrow$. – Chris Eagle Sep 4 '11 at 21:57
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I remember $\bot$ being used. See en.wikipedia.org/wiki/Partial_function#Bottom_type. – Srivatsan Sep 4 '11 at 21:59
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However $\not\exists f(x)$ seems confusing though. – Srivatsan Sep 4 '11 at 22:02
    
@Srivatsan You mean, $f(x) = \bot$? I remember something of the sort. – badp Sep 4 '11 at 22:09
    
Ah yes. I meant $f(x) = \bot$. (But this is just what I remember, so not to be trusted. Hopefully some expert can corroborate.) – Srivatsan Sep 4 '11 at 22:10
up vote 8 down vote accepted

A language for mathematical knowledge management uses $f(x)\uparrow$.

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I had not seen this before, but I have seen $f(x) \downarrow$ to mean that $f(x)$ is defined, so this makes sense. – Toby Bartels Jan 22 at 19:19

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