Question 1
Let $S$ be a set with $2002$ elements, and let $N$ be an integer with $0\le N\le 2^{2002}$. Prove that every subset of $S$ can be colored black or white so that:
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) exactly $N$ subsets are white.
Problem proposer credited in the source: Gabriel Carroll.
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) exactly $N$ subsets are white.
Problem proposer credited in the source: Gabriel Carroll.