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When learning $\LaTeX$, I saw some relation operators I haven't seen before, like $\nless$ and $\ngeq$. However, I don't see how those relation operators are different from the ones I already know. I'm talking about this pairs:

  • $>$ and $\nleq$
  • $<$ and $\ngeq$

Do the two operators in each of the above pairs mean exactly the same? If they are, what's the reason of having this symbols? If not, are they at least the same in $\mathbb{H}$, or else $\mathbb{C}$, or else $\mathbb{R}$? ($\mathbb{H} \subset \mathbb{C}$, right?)

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$\le$ can be used for an arbitrary ordering relation on an arbitrary set. When the ordering is not strictly total, $>$ is not necessarily $\nleq$.

For example, the ordering $\le$ on $\Bbb N$ given by $a \le b \iff a \mid b$ does not satisfy that.

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  • $\begingroup$ Thanks for your answer. What is an ordering relation? I'm also not familiar with the notation $a \le b \iff a \mid b$. What does it mean in words? $\endgroup$ – Kevin Jul 31 '16 at 16:59
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    $\begingroup$ @Kevin an ordering relation is a binary relation which is reflexive, antisymmetric and transitive. The notation $a \le b \iff a\mid b$ means that we are defining the relation $\le$ by: "$a \le b$ if and only if $a$ is a divisor of $b$". $\endgroup$ – user258700 Jul 31 '16 at 17:03
  • $\begingroup$ Thanks. Could you give me a concrete example of such a set? $\endgroup$ – Kevin Aug 7 '16 at 23:23
  • $\begingroup$ @Kevin you're welcome. What such set? Please precise the kind of set you are looking for. $\endgroup$ – user258700 Aug 7 '16 at 23:25
  • $\begingroup$ "an arbitrary set ... [where] the ordering is not strictly total" :) $\endgroup$ – Kevin Aug 7 '16 at 23:29

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