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# hw1f09solns[1] - 1 ISyE 3770 Fall 2009 Homework#1 Solutions...

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1 ISyE 3770 – Fall 2009 Homework #1 — Solutions 1. Suppose U = [0 , 2], A = [0 . 5 , 1], and B = [0 . 2 , 1 . 6]. What are A B , A B , A B , and A B ? Solution: Note that A B . Then A B = [0 , 0 . 2) (1 . 6 , 2] A B = [0 , 0 . 2) [0 . 5 , 1] (1 . 6 , 2] A B = [0 , 0 . 5) (1 , 2] A B = [0 . 2 , 0 . 5) (1 , 1 . 6] 2 2. Prove DeMorgan’s Laws. You can use Venn diagrams or argue mathematically. Solution: Let’s prove that A B = A B . Proof: x A B iff x / A B iff x / A and x / B iff x A and x B iff x A B 2 Now let’s prove that A B = A B . Proof: x A B iff x / A B iff x / A or x / B iff x A or x B iff x A B 2 3. A box contains 3 marbles (one red, one green, and one blue). (a) Consider an experiment that consists of taking one marble from the box, then replacing it in the box, and then drawing a second marble from the box. What is the sample space? Solution: S = { RR, RG, RB, GR, GG, GB, BR, BG, BB } 2 .

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