Consider the sample space $S = \{1, 2, 3, 4, 5, 6, 7, 8\}$ with equally likely outcomes. Let $A = \{2, 3, 5, 7\}$ (prime numbers) and $B = \{1, 3, 5, 7\}$ (odd numbers). What is the minimum number of elements a non-empty, proper subset $C \subset S$ can have such that $C$ is independent of both $A$ and $B$?