RecProbsTest2

RecProbsTest2 - Recommended problems 10/18/2005 CSCI 2670...

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10/18/2005 CSCI 2670 4.6 Assume B is countable and that f:N B is a functional correspondence. Let f i (j) denote the i th symbol in f(j) . Consider the string s whose i th element differs from f i (i) – i.e. if f i (i) = 0, then the i th symbol in s is 1 and vice versa. Then s cannot be in the image of f since it differs from every string in the image of f by at least one symbol. Furthermore, s is in B since it is an infinite sequence over {0,1}. Therefore, no functional correspondence can exist between N and B – i.e., B is uncountable. 4.7 We begin by lexicographically arranging the elements of T according to the sum of the elements {(1,1,1),(1,1,2),(1,2,1),(2,1,1),(1,1,3),(1,2,2),(1,3,1),(2,1,2), (2,2,1),(3,1,1),…}. For each natural number i , let f(i) be the i th element in the list. Since the list never repeats, f is clearly one-to-one. Also, for every element t in T, the sum of the components of t is finite. Therefore, there is some natural number n such that
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RecProbsTest2 - Recommended problems 10/18/2005 CSCI 2670...

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