# Is greater than ($>$) exactly the same as neither less nor equal to ($\nleq$)?

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?)

• In a totally ordered set yes, in a partially ordered set no. – Sri-Amirthan Theivendran Jul 31 '16 at 16:48
• – Steve Kass Jul 31 '16 at 16:49

$\le$ 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.
• 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? – Kevin Jul 31 '16 at 16:59
• @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 $b$". – user258700 Jul 31 '16 at 17:03