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?)
$\le$ 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.