How do partitions divide an event into mutually exclusive and exhaustive pieces?

"There’s a very useful fact that goes by the grandiose name “The Law of Total Probability.” It goes like this. If there’s an event whose probability we’d like to know, we can split it up into pieces and add up their probabilities, as long as we do it in the right way. “The right way” bit is the key, of course. And it has to do with partitions. Recall from section 2.12 that a partition of a set is a mutually exclusive and collectively exhaustive group of subsets. One example is that every set and its complement together form a partition of Ω. By the same token, for any sets A and B, these two sets together form a partition of A: A ∩ B and A ∩ Bᶜ. The first set contains members of A that are also in B, and the second set contains members of A that are not in B. Clearly, every member of A is in one of these two sets, and no member is in both. This works for any two sets A and B: A ∩ B and A ∩ Bᶜ are a partition of A. This idea can be extended to more than two sets. Let C₁ be the set of all people born in southern states, C₂ the set of people born in western states, and C₃ those not born in either region. The following three sets, then, together form another partition of A: A ∩ C₁, A ∩ C₂, and A ∩ C₃. This is because every professional wrestling fan is either born in the south, or born in the west, or neither one."