I feel like most definitions of “independence” are circular. Consider how we count the number of cards in a standard deck of cards: $|S times R| = |S||R|$, where $S$ is the set of suits, and $R$ is the set of ranks. That is, $$S = {text{Hearts, Diamonds, Spades, Clubs}}$$ and $$R = { 2,3,dots,text{King},text{Ace} }.$$ We know that $|S| = 4$ and $|R| = 13$, so $|S times R| = |S||R| = 4 cdot 13 = 52.$ In such an example, we define that they are independent, because they are disjoint subsets. But do we know that? In this example, it is “obvious.” What about examples where it is not obvious?