# Notation showing a set is non-empty and finite

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?

• You can write $S\ne \emptyset,|S|<\infty$ 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. Sep 23 '12 at 16:36

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 $\infty$

• @ZhenLin - corrected, thanks for pointing the typo out 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.

• So in that case could I also do $0 < |A| < inf$? Sep 23 '12 at 16:37
• @A.R.S. That's another way to put it. It would work too. 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. 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! Sep 23 '12 at 17:39
• @A.R.S. Use this: $\infty$. Sep 23 '12 at 18:06