TITLE: What is the meaning of the symbol $\not\geq$, and why would it be preferred to $<$?
QUESTION [8 upvotes]: I do not know if it is the right section to ask for it, but I wanted to ask some questions about math symbols that I often find on automation books. One of these is the following:
$$\not\ge$$
This symbol means no greater or equal, but what sense does it have? Why use this and not less "<" ?
Instead, this other symbol I think I do not know the exact meaning so I ask you for a confirmation: $$\gneq$$
Means greater than and at least one equal? Does it mean that if I basically compare two vectors, the first one must have elements greater than the second and must contain at least one equal?

REPLY [5 votes]: Without any context it is difficult to answer. 
Anyway generally speaking let's say that an order relation has been defined in a set whereby two elements of which may be in such a relation (and this order relation has been given this symbol $\ge$),
$a\ngeq b$ means either of the two:

$b \ge a$ and $a\ne b$
$a$ and $b$ are not comparable at all.

Instead $a\gneq b$ means both of the these two are satisfied:

$a \ge b$
$a \neq b$

Usually in a general context (that is, not with reals) when for an order relation a symbol like this $\ge$ is introduced that is like that used for "greater then or equal to" total order relation among reals, all the other symbols that are usually used with reals like these $<$, $>$, $\le$ are avoided, unless a definition for each is provided.