test1-fall90

# test1-fall90 - z || . PART D. Do TWO questions. (12 points)...

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TEST 1 MAS 3300 PART A. Do 2 questions. Prove (12 points) 1. The additive inverse is unique. (12 points) 2. - ( - a ) = a . (12 points) 3. 0 + 0 = 0. (12 points) 4. If a 6 = 0 then a a = 1. PART B. Do 2 questions. Prove (12 points) 5. If a > 0 and b > 0 then a + b > 0. (12 points) 6. If a > 0 and b > 0 then ab > 0. (18 points) 7. If a > 0 then a + 1 a 2. (24 points) 8. If x > 0 and y > 0 then x + y + 1 xy 3. (18 points) 9. If 0 < a < b then a < 2 1 a + 1 b < b . ( Hint: You may assume the result: if α < β then α < α + β 2 < β .) PART C. Do ONE question. Prove (12 points) 10. | a | ≥ 0. (18 points) 11. | a + b + c | ≤ | a | + | b | + | c | . (24 points) 12. | x - y - z | ≥ || x | - | y | - |
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Unformatted text preview: z || . PART D. Do TWO questions. (12 points) 13. State carefully: (a) The Well Ordering Principle and (b) The First Principle of Mathematical Induction. (12 points) 14. Use mathematical induction to prove For all integers n 1 we have 1 + 2 + + n = n ( n + 1) 2 . (18 points) 15. Use mathematical induction to prove For all integers n 1 we have | x 1 + x 2 + + x n | | x 1 | + | x 2 | + + | x n | , for all real numbers x 1 , x 2 ,..., x n . 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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