Quiz2_sol

# Quiz2_sol - A has a least element That is there exists an...

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QUIZ 2 WED., OCT. 6 NAME: DIRECTIONS: No papers, phones, calculators, or gadgets are permitted to be out during the quiz. Show all work, clearly and in order You will lose points if any of these instructions are not followed. Questions Points Score 1 1.5 2 1.5 3 2 Total 5 1

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QUIZ 2 WED., OCT. 6 NAME: Problem 1: (1.5 point) Let A i be sets for i = 1 , 2 , 3 , ..., n . What is the deFnition of the n-fold Cartesian Product of sets A 1 , A 2 ,..., A n ? The n-fold Cartesian product of sets A 1 ,..., A n is given by A 1 × A 2 × · · · × A n = { ( a 1 , a 2 , ..., a n ) | a j A j for j = 1 , 2 , ..., n } . Problem 2: (1.5 point) State the Well-Ordering Principle for Z . If A is a non-empty subset of the positive integers, then
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Unformatted text preview: A has a least element. That is, there exists an element a ∈ A such that for all a ∈ A , a < a . Problem 3: (2 points) Consider the set of integers, Z . Prove that the multiplicative identity is unique. Proof: Suppose 1 and 1 ′ are both multiplicative identities. Therefore, for all a ∈ Z (1) 1 · a = a · 1 = a, and (2) 1 ′ · a = a · 1 ′ = a. In particular (1) holds for a = 1 ′ and (2) holds for a = 1 hence 1 = 1 · 1 ′ = 1 ′ · 1 = 1 ′ . Therefore the multiplicative identity must be unique. 2...
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Quiz2_sol - A has a least element That is there exists an...

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