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hw1 fall 07 solns

# hw1 fall 07 solns - 1 ISyE 3770 – Fall 2007 Homework#1...

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Unformatted text preview: 1 ISyE 3770 – Fall 2007 Homework #1 — Solutions 1. Suppose U = [0 , 2], A = [0 . 5 , 1], and B = [0 . 25 , 1 . 5]. What are A ∪ B , A ∪ B , A ∩ B , and A ∩ B ? Solution: Note that A ⊆ B . Then A ∪ B = [0 , 1 4 ) ∪ ( 3 2 , 2] A ∪ B = [0 , 1 4 ) ∪ [ 1 2 , 1] ∪ ( 3 2 , 2] A ∩ B = [0 , 1 2 ) ∪ (1 , 2] A ∩ B = [ 1 4 , 1 2 ) ∪ (1 , 3 2 ] 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?is the sample space?...
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hw1 fall 07 solns - 1 ISyE 3770 – Fall 2007 Homework#1...

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