I've seen someone asking a question with $\gneq$ ($\gneq$) in it. What does it mean? What's the difference with $\geq$ ($\geq$)?
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1$\begingroup$ "gneq" means "greater than not equal". For the meaning, we will need to see some context. If it is in a question, then there is a natural place to get clarification, right? $\endgroup$– GEdgarCommented Dec 8, 2011 at 15:26
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1$\begingroup$ But I am waiting to hear for someone, that is the question correctly posed ? , or there is a need for some edit ? $\endgroup$– IDOKCommented Dec 8, 2011 at 15:28
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$\begingroup$ @GEdgar : But we can simply use "greater than sign" which means that they are not equal implicitly $\endgroup$– IDOKCommented Dec 8, 2011 at 15:29
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2$\begingroup$ @iyengar:The title seems correct to me and the difference is only the second part of the question. $\endgroup$– QuixoticCommented Dec 8, 2011 at 15:31
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1$\begingroup$ @iyengar: The question is fine. The OP has asked for the distinction between two symbols. You think they should ask about other symbols; that is your opinion. $\endgroup$– Zev ChonolesCommented Dec 8, 2011 at 15:34
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2 Answers
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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 $\subseteq$.
answered Dec 8, 2011 at 15:28
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$\begingroup$ This is clearly the general usage. It might be worthwile to point out that the set-theoretic notation $\subset$ is not universally used for proper inclusion. To be on the safe side, most people only use $\subseteq$ and $\subsetneq$. $\endgroup$– RasmusCommented Dec 8, 2011 at 15:30
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$\begingroup$ @iyengar: I don't know. That's why I said may. $\endgroup$ Commented Dec 8, 2011 at 15:32
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$\begingroup$ @Rasmus: Thanks for the suggestion, I've included that (ha!) $\endgroup$ Commented Dec 8, 2011 at 15:38
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2$\begingroup$ @iyengar: If one says "A means the same as B", where B is widely considered to have a different meaning than C, then one has explained the difference between A and C. $\endgroup$ Commented Dec 8, 2011 at 15:55
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$ 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.
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$\begingroup$ "good try" -??? That sounds diminutive. iyengar, there's no difference. $\endgroup$ Commented Dec 8, 2011 at 17:17
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$\begingroup$ yes we can write $x^2+1 \gt 0$ but when we write $x^2+1 \gneq 0$ we emphasis that it can't be zero. I mean the point is emphasizing because I've seen this sign when it was necessary to not be equal. For example when the statement is in the denominator of a fraction and we want to emphasis that it can't be zero. And Thanks for the +1! $\endgroup$– BardiaCommented Dec 8, 2011 at 17:23
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$\begingroup$ @Bardia : Very nice answer !! , convinced with it, why can't you include the same application of fractions in your answer $\endgroup$– IDOKCommented Dec 8, 2011 at 17:30
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$\begingroup$ @TheChaz : dimunitive ? , I never know the person before, but just wanted to tell so, but if you mind keeping that I better delete it $\endgroup$– IDOKCommented Dec 8, 2011 at 17:33
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$\begingroup$ @iyengar: fair enough! I should have included that example too. $\endgroup$– BardiaCommented Dec 8, 2011 at 17:42