Worked-Probability-Problems

Worked-Probability-Problems - ARE 210 Worked Examples in...

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ARE 210 Worked Examples in Probability Fall 2007 1. Prove A (B C) = (A B) (A C) By the definition of the equality of sets, A (B C) = (A B) (A C) if A (B C) (A B) (A C) and A (B C) (A B) (A C). That is, we must show that any element of the set A (B C) must also be an element of (A B) (A C) and vice versa. If x A (B C), then x A and x B or x C. But this is the same as x A and x B or x A and x C. Hence, x (A B) (A C) which means that A (B C) (A B) (A C). Conversely, let x (A B) (A C). Then x (A B) or x (A C). Since x A either way and x B or x C, we have x A (B C). As a result, we have shown that A (B C) (A B) (A C). 2. Prove A (B C) = (A B) (A C) Suppose x A (B C). Then x A or x (B C) so x A or x B and x C. If x A, then x (A B) and x (A C). If x B and x C then it is also true that x (A B) and x (A C). In either case, x (A B) (A C) so A (B C) (A B) (A C). To show the other direction, suppose x (A B) (A C). Then, x (A B) and x (A C). Note that if x A, then x must be an element of both B and C so x A (B C). Therefore, A (B C) (A B) (A C). 3. Prove () c c ii AA ∩= Let c i A i x ∈∩ . Then i A i x ∉∩ , or equivalently, i x A for some i. Since x A i for some i , it must be true that x
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This note was uploaded on 08/01/2008 for the course ARE 210 taught by Professor Lafrance during the Fall '07 term at University of California, Berkeley.

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Worked-Probability-Problems - ARE 210 Worked Examples in...

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