Given N possibly overlapping sets, give formulas that specify, using intersections and complements of the given sets, N disjoint sets with the same union as the original N sets. The sets that result are to be the same as the given sets in the case where the given sets are already disjoint.

If we have sets A, B, C, ... Y, Z, a solution is

A' = A - B - C - ... - Y - Z

B' = B - C - ... - Y - Z

C' = C - ... - Y - Z

...

Y' = Y - Z

Z' = Z

If you want to use just intersections and complements, apply the fact that P-Q = P intersection (complement of Q).