What is $\gneq$?

Question

I've seen someone asking a question with $\gneq$ ($\gneq$) in it. What does it mean? What's the difference with $\geq$ ($\geq$)?

Answer 1

I would think $\gneq$ means exactly the same as $>$, i.e. it would mean greater than and not equal to (while the symbol $\geq$ means greater than or equal to). But of course there may be some specialized use where it doesn't mean this though; everything depends on context.

In the context of the question you linked to, I can say with certainty that the intended meaning is the one above. That is,

$$n\gneq 3 \iff n>3 \iff n\text{ is greater than }3$$ and, because $n$ is an integer in this context, we can also say that $$n\gneq 3\iff n\geq 4.$$

As Rasmus points out below, the analogous notations with set inclusion, $\subset$ vs. $\subsetneq$, unfortunately do not mean the same in general; many authors use $A \subset B$ to mean "$A$ is a subset of $B$, and could be equal to $B$". An unambiguous alternative to express that would be to write $\subseteq$.

Answer 2

$ a \geq b$ means that $a$ is greater than $b$ or it can be equal to $b$.

$a \gneq b$ means $a$ is greater than $b$ and it can't be equal to $b$.

The $\gneq$ sign used when we want to emphasis that they can't be eqaul.

for example I can write $x^2 +1 \geq 0$ and it is true because it means $x^2 +1$ is greater than zero or it can be equal to zero. (I hope you remember how the or operator works.)

but it is better to say that $x^2 +1 \gneq 0$ which means $x^2 +1$ is greater than zero and it can't be zero.