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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SteeveDroz
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1"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? – GEdgar Dec 08 '11 at 15:26
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1But I am waiting to hear for someone, that is the question correctly posed ? , or there is a need for some edit ? – IDOK Dec 08 '11 at 15:28
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@GEdgar : But we can simply use "greater than sign" which means that they are not equal implicitly – IDOK Dec 08 '11 at 15:29
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2@iyengar:The title seems correct to me and the difference is only the second part of the question. – Quixotic Dec 08 '11 at 15:31
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@MaX : That's what I told – IDOK Dec 08 '11 at 15:32
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1@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. – Zev Chonoles Dec 08 '11 at 15:34
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It's been a long time, but I vaguely recall that when talking about partially ordered sets and lattices and the like, one often sees $<$ with the condition $a<a$, so that $<$ will sometimes "really" be $leq$. This is similar to the existence of notations $\subset$, $\subseteq$, and $\subsetneq$ – Thomas Andrews Dec 08 '11 at 15:35
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@ZevChonoles : Then According to OP question it means that he is asking the difference between $A\gneq B$ and $A\ge B$ , we can surely tell the difference the former one allows $A$ not equal to $B$ but latter one allows $A$ to be equal to $B$, what do you say? – IDOK Dec 08 '11 at 15:36
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But I think the question should be "What's the difference between $\gneq$ and $\gt$ ? , as if $A\gneq B$ it implicitly means that $A\gt B$, but what's the need of using the different symbols then, we can directly use $A \gt B$ as it means implicitly that "$A $ is greater than $B$, which implies that they are not equal anymore, so I think that $\gneq$ and $\gt$ mean the same, Don't they ? – IDOK Dec 08 '11 at 15:40
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We need the context. When writing $A \gneq B$, what are $A,B$? Maybe there is some meaning defined for rather frugal loopoids $A,B$ and $A \gneq B$ and $A \ge B$ have different meanings... So we must await the return of Oltarus to find the source. – GEdgar Dec 08 '11 at 15:59
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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 $\subseteq$.
Zev Chonoles
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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 $\subsetneq$. – Rasmus Dec 08 '11 at 15:30
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@ZevChonoles : you suggested me that question was fine, but you look at it once, the OP asked the difference between $\gneq$ and $\ge$, but your answer was telling the difference between $\gneq$ and $\gt$, thats why I have been saying that question should be edited, and you said its "fine". – IDOK Dec 08 '11 at 15:43
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@iyengar: My answer is not telling the difference between $\gneq$ and $>$, it was explaining the meaning of $\gneq$ by claiming it is interchangeable with $>$. At any rate I have now included a comment contrasting this with $\geq$. – Zev Chonoles Dec 08 '11 at 15:48
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2@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. – Zev Chonoles Dec 08 '11 at 15:55
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@ZevChonoles : "OP asked that what is the difference between $\gneq$ and $\ge$, but your answer was primarily explaining the similarity between $\gneq$ and $gt$ ( recently you added something in the answer present in brackets ) , but the point I was stressing was that "OP might have thought to ask the difference between the $\gneq$ and $gt$ but used "$\ge$ " instead of "$\gt"$, if the question is re-edited it would make much sense, but the question present in the above form is universally known as everyone knows the difference between $\gneq$ and $ge$. You understood my point ? – IDOK Dec 08 '11 at 16:02
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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.
Bardia
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"good try" -??? That sounds diminutive. iyengar, there's no difference. – The Chaz 2.0 Dec 08 '11 at 17:17
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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! – Bardia Dec 08 '11 at 17:23
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@Bardia : Very nice answer !! , convinced with it, why can't you include the same application of fractions in your answer – IDOK Dec 08 '11 at 17:30
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@TheChaz : dimunitive ? , I never know the person before, but just wanted to tell so, but if you mind keeping that I better delete it – IDOK Dec 08 '11 at 17:33
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@Bardia : do it now ! , nice example, I was waiting for such practical application, really practical situation to contrast their usage, I searched in google for much time, but didn't found any such examples or situations. – IDOK Dec 08 '11 at 17:44