test1-summer01 - then-a>-b 6 If a ∈ R and 0< a< 1...

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Name: CODEWORD: MAS 3300 – Test 1 Summer A 2001 Give proofs of the following quoting the relevant axioms and results. Proofs should be written in a proper and coherent manner. Write so that anyone in the class can follow your work. The questions have equal value. Write on ONE side of your paper. 1. The additive identity is unique. That is, if e R and a + e = a = e + a for all a R , then e = 0. 2. If a R then a 0 = 0 . 3. If a R , then - a = ( - 1) a. 4. Suppose a , b , c , and d R . If b 6 = 0 and d 6 = 0, then a b + c d = ad + bc bd . 5. Suppose a and b R . If a < b
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Unformatted text preview: , then-a >-b . 6. If a ∈ R and 0 < a < 1 then 1 a < 1 a 2 . 7. If a ∈ R then | a | = | -a | . 8. If a ∈ R and 0 < a < 1, then a 6∈ N . You may not assume Theorem 2.3 (2). (Hint: Suppose by way of contradiction that a ∈ N . Let A = { m ∈ N : 0 < m < 1 } . Now use the Well Ordering Principle and Theorem 2.3(1).) 9. For each positive integer n , 1 + 2 2 + 3 2 + ··· + n 2 = n ( n + 1)(2 n + 1) 6 . 1...
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