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Does there exist any mathematical notation that would indicate that a set $V$ is non-empty and finite? Or would I have to write this out in words?

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You can write $S\ne \emptyset,|S|<\infty$ – pritam Sep 23 '12 at 16:34
For some additional suggestions for finite, see this link (one can easily add in the non-empty part). Best not to do it, usually. Words are good. – André Nicolas Sep 23 '12 at 16:36
up vote 4 down vote accepted

To say that $V$ is not empty you can either say so or write $V\neq\emptyset$ or $|V|>0$.

To say that $V$ is finite you can either say so or write $|V|<\aleph_{0}$.

So you can write something like $0<|V|<\aleph_{0}$ to say that $V$ is a non-empty finite set.

Added: in many context (mainly non set theory wise where just writing infinity is not common) you can replace $\aleph_{0}$ with $\infty$

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@ZhenLin - corrected, thanks for pointing the typo out – Belgi Sep 23 '12 at 16:39

You can use the cardinality notation. The cardinality of a set $A$ is usually denoted as $|A|$. If the set is non-empty and finite, you can express this as:

$$ A \neq \emptyset, |A| < \infty $$

However, I think that explaining this in words would be clearer.

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So in that case could I also do $0 < |A| < inf$? – arshajii Sep 23 '12 at 16:37
@A.R.S. That's another way to put it. It would work too. – Ayman Hourieh Sep 23 '12 at 16:38
@A.R.S. : I wouldn't use inf, $\infty$ is really more appropriate. The reason for this is that the word inf is mostly used for infimums in analysis, so your last sentence is just as readable as $a+\% b=! ?$ for some people. It would be understood but not at first sight. – Patrick Da Silva Sep 23 '12 at 17:08
@PatrickDaSilva Yea you're right - I just didn't know how to obtain the infinity symbol on the comment :P But thank you for the tip! – arshajii Sep 23 '12 at 17:39
@A.R.S. Use this: $\infty$. – Ayman Hourieh Sep 23 '12 at 18:06

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