# What is the meaning of the symbol $\not\geq$, and why would it be preferred to $<$?

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?

• In some contexts, you're not necessarily allowed to compare any two elements. Using $\supseteq$, set containment, instead of $\geq$, it's a bit clearer that $\not\supseteq$ isn't the same as $\subset$. – Arthur Jul 22 '17 at 8:02
• To elaborate on @Arthur's comment, the binary relation on a set $X$ to which $\leq$ and $\geq$ refer is usually called an order. The usual ordering on the real numbers is an example of a total order, which comes with the axiom $x \leq y$ or $y \leq x$ for all $x, y \in X$. But some orders do not come with this totality axiom, like partial orders, wherein two elements might not be comparable at all, so $x \not\le y$ would not imply $x>y$: en.wikipedia.org/wiki/Partially_ordered_set#Formal_definition – Kaj Hansen Jul 22 '17 at 8:34

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:

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

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

1. $a \ge b$
2. $a \neq b$

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.

• An example of the use of $\ngeq$ could be if we had a set of letters in the Roman alphabet together with the integers. Then $1\ngeq 5$ (and it is true that $1<5$), while $\mathrm{G} \ngeq 1$ (but it is not true that $1 < \mathrm{G}$). – dbmag9 Jul 22 '17 at 11:02

$a\ge b$ means $a>b$ or $a=b$. And $a\not\ge b$ is the negation thereof, i.e., neither $a>b$ nor $a=b$. If we were talking about a total order, this would be equivalent to $a<b$. But maybe we are not.

Mathematical Symbols are there to express mathematical ideas in a short, precise and understandable way. If you want to express that "a is not greater than or equal to b", it could be argued that $a \ngeq b$ is the most direct expression for that.

The other symbol, $a \gneq b$ means "a is greater but NOT(!) equal to b". The crossed out line of the equal sign can be understood as an emphasis on the "not equal" part. From a purely logical standpoint, $a \gneq b$ and $a > b$ mean exactly the same.

I think they used $$\not\ge$$ instead < to tell readers about to write less than in other form. Because most of the time in exams questions are not tough but in order to confuse student they write symbols in different form.

Example - a is less than b. Its easy to interpret. But a is neither greater than nor equal to b makes some confusion.

$$\gneq$$ means its greater but not equal.