MAT 215 Quiz

MAT 215 Quiz - sions s = 0 .d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8...

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Quiz — Math 215 Fall 2008 You may use Rudin for the quiz. In your answers, you may cite results proven in Rudin. No other source may be consulted. You may not discuss the quiz with anyone. 75 minutes, 3 questions I. Let Q R be the subset of rational numbers in the real numbers with the standard metric. (i) A closed interval is a subset of R of the form I a,b = { r R | a r b } for real numbers a < b . If C, D R are closed intervals and C Q = D Q , must C = D ? (ii) If U, V R are open sets and U Q = V Q , must U = V ? Answer all questions with proof. II. Which of the following sets are countable? Answer with proof (that Q is countable and R is uncountable need not be proved). (i) The subset X R 2 de±ned by X = { ( a, b ) | a R , b R , a + b Q } . (ii) The subset Y R 2 de±ned by Y = { ( a, b ) | a R , b R , a + b Q , ab Q } . 1
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2 (iii) The subset S R of real numbers 0 < s < 1 with decimal expan-
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Unformatted text preview: sions s = 0 .d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 . . . with digits satisfying d i d i +1 for all i . For example, . 97777665555555555555555555555555 . .. ending in innitely many 5s is in S , but / 10 = 0 . 314159265358979323846 . . . is not in S . III. Suppose E R is a compact set with a countable (innite) number of elements. For each question below, provide an example or prove the impossibility. (i) Can the set of limit points of E in R be empty? (ii) Can the set of limit points of E in R have exactly 5 elements? (iii) Can the set of limit points of E in R contain the set Q (0 , 1) ? (iv) Can the set of limit points of E in R contain all the integers Z ?...
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MAT 215 Quiz - sions s = 0 .d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8...

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