test1-spring97 - a , b , c ∈ R . Prove | a-c | ≤ | a-b...

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TEST 1 - MAS 3300 - Spring 1997 February 14, 1997 There are 7 questions giving a total of 45 points. Answer all questions. Work submitted must be written in a proper and coherent fashion. Write on ONE side of your paper. 1 . [5 pts] Let a R . Prove a 0 = 0 , stating the necessary axioms and results. 2 . [5 pts] Let a R . Prove that the additive inverse of a is unique. Be sure to state the necessary axioms and results in your proof. 3 . [5 pts] Let a and b R . Prove that if a < b , then - a < - b . Be sure to state the necessary order axioms and results in your proof. 4 . [2 + 5 = 7 pts] (i) State the Triangle Inequality . (ii) Let
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Unformatted text preview: a , b , c ∈ R . Prove | a-c | ≤ | a-b | + | b-c | . 5 . [2 + 6 = 8 pts] (i) State the First Principle of Mathematical Induction . (ii) Prove 1 · 2 + 2 · 3 + 3 · 4 + ··· + n ( n + 1) = 1 3 n ( n + 1)( n + 2) , for all integers n ≥ 1. 6 . [2 + 5 = 7 pts] (i) Define the term ideal . (ii) Let a ∈ Z . Prove that the set J a := { ka : k ∈ Z } is an ideal of Z . 7 . [2 + 6 = 8 pts] (i) Define what it means for two integers to be relatively prime . (ii) Let a , b ∈ Z . Prove that gcd( a,b ) = 1 if and only if gcd( a + b,ab ) = 1. 1...
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This note was uploaded on 06/03/2011 for the course MAS 3300 taught by Professor Staff during the Spring '08 term at University of Florida.

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