Is $\nless$ standard notation in Mathematics

Question

I was recently told that $\nless$ symbol doesn't exist. I know $\nless$ is equivalent to $\geq$, but i wondered if $\nless$ exists, or rather, is used in some theorem or math paper or def, etc. Something to have a reference that it actually exists (or that it doesn't).

Answer 1

The symbol does exist and is used. In terms of real numbers, $\not <$ is equivalent to $\ge$, but this is not true for more general orders on sets like partial orders.

A good introductory example of a partial order is that $x<y$ for natural numbers $x,y$ if and only if $x$ is a proper factor of $y$. Here, it may be a useful notation. For instance, the true statement $4\not < 6$ ("$4$ is not a factor of $6$") is not equivalent to the false statement $6\le 4$ ("$6$ is a proper factor of $4$, or equal to $4$").

This is a somewhat contrived example, as we would likely use a different symbol to $<$ for any partial order involving natural numbers, to avoid confusion with the "normal" $<$ (and for divisibility we often use $|$ as a symbol). However, we would use $<$ when discussing partial orders in general, or where there is no "standard" usage of $<$ that already applies.

The only case where I would use $\not <$ for cases involving real numbers is informally, for emphasis. For instance, if someone had written $2.4<2.3$ I might write "the mistake here is that $2.4\not <2.3$" or something like that.