Question 169
Let $X$ and $Y$ be two finite subsets of the half-open interval $[0, 1)$ such that $0 \in X \cap Y$ and $x + y \neq 1$ for all $x \in X$ and $y \in Y$. Prove that the set $\{x + y - \lfloor x + y \rfloor \mid x \in X \text{ and } y \in Y\}$ has at least $|X| + |Y| - 1$ elements.