What is the purpose of the $\mp$ symbol in mathematical usage? Occasionally I see the $\mp$ symbol, but I don't really know what it is for, except in conjunction with the $\pm$ symbol thus: $a \pm b \mp c$ which (I believe) means $a+b-c$  or  $a-b+c$ (please correct me if I am wrong). Is there any other mathematical usage for the $\mp$ symbol, particularly on its own ?
 A: You are correct; $\mp$ only makes sense in a formula that already has $\pm$.
One simple and useful example is that when $x$ is small, ${1\over{1\pm x}}\approx 1\mp x$.
A: Like the other answerer, I've only seen it used in the same line as a $\pm$, to mean "positive when the other term is negative and negative when the other term is positive." So, for instance, if we were to say
$\pm a = \mp b$
that would imply that
$ a = -b $
and
$ -a = b $
A: $\mp$ really only has a use when written in the same expressions as $\pm$.
The one that comes to mind is $\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$. 
But I suppose if you really wanted to, you could write things like $\sin(\alpha \mp \beta) = \sin \alpha \cos \beta \mp \cos \alpha \sin \beta$... if you really wanted to.
On a more humorous vein, it wouldn't surprise me if someone overloaded the symbol to have a different meaning too. Most likely someone like Conway (as in, Combinatorial Game Theory Conway, not Complex Analysis Conway), who thought $+_n$ was a perfectly good name for a state of a game (not an operation).
an aside
On a non-mathematical note, $\pm$ denotes an advantageous position for white in chess. $\mp$ denotes a position for black.
If we really go for it, $\mp$ looks like (干) wiki page, which means 'to dry' in Japanese and might mean 'to do' in Mandarin. $\pm$ looks like (士)wiki page, which might mean 'gentleman' in Japanese and is used in the symbols for doctorate and doctor's thesis.
