As you have rightly observed you are having problems because \c is already provided. Its purpose is to provide the cedilla diacritic, which can actually be added to a number of letters. The letter it is to be added to should appear as an argument of \c, e.g. \c{c} produces ç.
Accordingly, if you want to define an alternative \c macro, you must issue \renewcommand like so:
\renewcommand{\c}{\mathrm{c}}
Your code will now work.
This, however, is not necessarily a very good idea. If you ever need to write about limaçons you may regret doing this. Steven B. Segletes provides a superb compromise for this case. All the same, conventional wisdom is to be very careful about redefining basic LaTeX commands, unless you're confident you know what you're doing with them. Otherwise you may end up breaking things. The weirdest error I ever got was when I thought I'd be very clever and write a little package called trig, to provide commands like \arccsc. Little did I know that there actually already exists a trig package, which is used for some definitions required in the pdflscape package. Oh I had fun when I tried to render a document with one landscape page.
Anyway, even if you don't need \c, by redefining basic commands, you make your code unlikely to work well for anybody else. This is fine if you're just doing your maths homework, but beware of getting too used to your definition of \c if you then collaborate with someone else. You may also find that some editors' syntax highlighting will go awry with some redefinitions.
A nicer alternative is to simply use a similar definition which isn't already defined.
I recommended
\newcommand{\myc}{\mathrm{c}}
Which will give you no such problems, but it doesn't matter if this isn't to your taste. Anything you fancy. But you will have fewer problems if you're cautious about overwriting LaTeX's definitions.
It's your code, though, just maybe add a comment to remind yourself if you do redefine \c?
An alternative that may help other people is to think about finding an editor with good autocomplete functionality, or which can input long commands if you find them too cumbersome. Have a look through this list and see what might work for you. I now know that you yourself are not so concerned about input but others may find this a better alternative.
\renewcommand, but in the words of egreg, never redefine a command you don't fully understand. Could you not define a command that doesn't already exist. Something like\myc. Another alternative, of course, is to get an editor with good autocomplete functions, so that you don't have to define lots of new short commands to save yourself the typing. This is generally better as you don't get these problems and others can use your code safely, without the risk that your new definitions will break something or clash – Au101 Jun 17 '15 at 00:13\complementevery time. Andmycis just a very bad name for a command. If the "c-cedilla" is so fundamental to LaTeX that one cannot deactivate it, I guess I better switch to using justcwithout\mathrm. Thanks anyway. – Andrey Tyukin Jun 17 '15 at 00:19\cis already defined, as a macro that puts a cedilla below its argument. Hence, the instruction\providecommand{\c}{\mathrm{c}}won't do anything. You need\renewcommandinstead of\providecommand. Of course, redefining\cwill make it impossible to use the macro to insert cedillas elsewhere in your document. – Mico Jun 17 '15 at 00:22\renewcommandand\providecommand(I somehow got the wrong idea that\providecommandeither creates or overrides the previous definition). – Andrey Tyukin Jun 17 '15 at 00:32A^\complementand then may be redefine\complement(I usually use a\mathsf{c}) or even change the definition to\complement{A}so you can change the output completely at the end. – Manuel Jun 17 '15 at 10:16