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I have two vectors, $x, y\in \mathbb{R}^N$ with $N>1$. Consider an order such that $x\geq y$ iff $x_i\geq y_i$ for each $i=1,2,\ldots, N$, and $x> y$ iff $x_i> y_i$ for each $i=1,2,\ldots, N$. I want to highlight the fact that

(a) $x\geq y$,

(b) $x\neq y$, and

(c) $x\not> y$.

Can I use $x\gneqq y$? It seems natural to me but I do not want to confuse readers. Unfortunately, my Google searches seems to show that there is no consensus on how to use $\gneqq$.

Note that $N>1$ is important here. When $N=1$, $x\geq y$ and $x\not>y$ immediately imply $x=y$.

tsm
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    If $x\geq y$ and $x\not >y$, then $x=y$. All three (a), (b), and (c) cannot occur simultaneously. The notation $x \not >y$ usually means "$x$ is not greater than $y$" to most mathematicians. But yes, $\gneqq $ can be used to emphasize unequalness. – Batominovski Jul 08 '18 at 01:16
  • I do not understand why the $\ne$ symbol is needed. Unless you have some special ordering in mind I presume that you are speaking of the relative magnitudes of the vectors. In that case, if the magnitudes are different, the vectors are different and there is no need for the $\ne$ symbol. – John Wayland Bales Jul 08 '18 at 01:26
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    So take $x,y\in \mathbb{R}^2$. I want to highlight situations such as $x_1=y_1$ but $x_2>y_2$. In this case, $x\geq y$ by the standard ordering. It is NOT the case that $x=y$ or $x>y$ (which would require $x_1>y_1$ and $x_2>y_2$). – tsm Jul 08 '18 at 01:58
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    I have seen $\gneqq$ used for precisely this purpose before and I think it’s reasonably intuitive. Just make sure to introduce it when it occurs for the first time. – Eike Schulte Jul 08 '18 at 10:02
  • Sorry, but what is the ordering here? What does $x>y$ even mean for two vectors $x,y\in \mathbb R^2$? Are you trying to say $|x| > |y|$? – mweiss Jul 08 '18 at 18:30
  • @tsm I don't know what you mean by "standard ordering" but I am not aware of any ordering in which $x_1=y_1, x_2 > y_2$ means $x \ge y$ but at the same time $x > y$ requires both $x_1>y_1$ and $x_2 > y_2$. – mweiss Jul 08 '18 at 18:47
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    I think you have the lexicographic order in mind, but have the definition mixed up. In any partial or total order, the symbol $x > y$ means precisely that $ x \ge y$ and $x \ne y$. – mweiss Jul 08 '18 at 18:49
  • I am defining $x\geq y$ as $x_i\geq y_i$ for each component $i=1,2,\ldots, N$. You are right: $x>y$ iff $x\geq y$ and $\neg ( y\geq x)$. I believe the confusion arises (from lack of my clarity) because I was defining $x>y$ as $x_i>y_i$ for each $i$. – tsm Jul 08 '18 at 19:10
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    I think $x\gneqq y$ would suggest $x>y$ and $x\ne y$, which is not what you want to convey. I would just write something like $x\ge y$ and "note that $x\ne y$ and $x\not> y$". – Mark S. Jul 09 '18 at 01:48

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