M413finalsol - Solutions for Final Exam 1 State the field...

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Solutions for Final Exam 1 . State the field axioms of the real number system: See the book or my notes. Several students made the minor error of forgetting that an element must be nonzero to have an inverse. 2 . State the Completeness Axiom of the real numbers. A nonempty bounded set of real numbers has a least upper bound. 3 . Use the Rational Zeros Theorem to show that the number b = 3 - 2 is irrational. We have b 2 = 3 - 2 so b 2 - 3 = - 2 and ( b 2 - 3) 2 = 2. Therefore, b is a solution of the equation b 4 - 6 b 2 - 7 = 0. By the theorem, the only possible rational solutions are b = ± 1 1 , ± 7 1 or b = 1 , - 1 , 7 , - 7. But none of these work, so we must conclude b is irrational. 4 . Find the supremum and the infimum of each set: a. [0 , 5) sup[0 , 5) = 5 and inf[0 , 5) = 0. b. 1 2 n + 1 : n N This set is 1 3 , 1 5 , 1 7 , 1 9 , . . . , so the supremum is 1 / 3 and the infimum is 0. 5 . Suppose S is a nonempty set of real numbers. Suppose m = max S exists. Prove that sup S = m . We have that m is an upper bound for S since x m for all x S . If n is any other upper bound for S , we have m n since m S . Therefore, m is the least upper bound for S . That is, m = sup S .
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