Worked-Probability-Problems

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 5

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online