Do you mean an arithmetical union ("+") or logical conjunction ("^")?
If the latter, do you mean:
C ^ C ~C = (C v C) ^ ?
This would be simpler if you could tell us whether you mean for "C" as a algebraic variable or as a symbol, for example the first letter in someone's last name.
And if the first letter, does each "C" represent the same person or is each C meant to describe a different person? In which case do you mean:
C1 ^ C2 ~(C1 v C2) = (C1 v C2) ^ ?
For God's sake, tell us what you mean -- my head hurts. But not as much as if your conjecture is the latter not the former and turns out true. That would be pretty sad, indeed.