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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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 Fall '07
 SCHMIDT
 Natural number, Prime number, multiplicative identity, sets A1

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