I'm writing a academic paper and need to represent "A is not the empty set". What is usual way for professional mathematicians?
My idea is:
$|A| > 0$
However, using the emptyset $\emptyset$ might be more intuitive:
$A\ != \emptyset$
But, I know "!=" is not permissible in math community (only in programmers).
Update
Sorry, I fixed the second equation: $|A|\ != \emptyset \rightarrow A\ != \emptyset$
6 Answers
It is perfectly fine to write $|A|>0$. However, the simplest and most common way to write this in symbols would be $$A\neq\emptyset.$$ Note that you don't want to write $|A|\neq \emptyset$, as it is $A$ itself which you are saying is not the empty set, rather than the cardinality of $A$.
(The standard symbol in mathematics for "not equal" is $\neq$, rather than $!{=}$. You can make this symbol in $\LaTeX$ with the command \neq.)
As mentioned in user21820's nice answer below, though, it is also very common to just write this in words ("$A$ is not empty" or "$A$ is nonempty") instead of symbols.
None of the other answers so far mention that professional mathematicians don't specially go out of the way to convert everything to symbols. "$A$ is non-empty" is indeed the most common way to express the statement. Furthermore, for complicated structures it is almost always expressed this way, such as:
Given any non-empty chain of fields ordered by inclusion, their union is also a field.
$A \neq \varnothing$
[LaTeX: A \neq \varnothing]
$A \neq \emptyset$
Thats how it is commonly written
How about: $(\exists x)\, x\in A$? Alternatively, one could say that $A$ is inhabited. This usage avoids needless negation which is problematic constructively speaking, and is common in constructive mathematics.
Let $A$ denote a set.
In constructive mathematics, there's a difference between the statements '$A$ is non-empty,' which is defined to mean that $A$ is not isomorphic to $\emptyset$, and '$A$ is inhabited,' which is defined to mean that $A$ has at least one element, i.e. $\exists a \in A(\mathrm{True})$. Thus, depending on their standpoint and interests, a professional mathematician will probably write one of '$A$ is non-empty' or '$A$ is inhabited.'
Classically, these are equivalent.
