hw5-solutions

# Then it is possible that the real number m appears on

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Then it is possible that the real number M appears on our list. Provide a list of real numbers for which this happens. (Recall that 0 . 1999 . . . = 0 . 2.) (B+S 3.3.21; modified a bit because the presentation of Cantor’s proof in the text is different from the one done in class) Solution We want a list of real numbers which has 2s down the diagonal (except for perhaps at the beginning); then this process will generate a real number ending in infinitely many 9s, thus making the ambiguity possible. So we construct a sequence for which this process will give . 2999 . . . , and make sure . 3 (which equals . 2999 . . . ) is on our list. One possible list is r 1 = . 30000 . . . r 2 = . 22000 . . . r 3 = . 22200 . . . r 4 = . 22220 . . . r 5 = . 22222 . . . where the n th number on the list, r n , consists of n 2s and then infinitely many zeroes. 1

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3. Suppose that Words is the set defined by Words = { all, you, infinity, found, them, search, the, it } . Consider the following pairing of elements of Words with elements of P(Words): Elements of Words Elements of P(Words) all { all, infinity, found } you { it, search, them } infinity { all, them, infinity } found { you, the, it } them { found, them } search { all, infinity, search } the { the, search } it { infinity, you, all } Using the idea of Cantor’s proof of the uncountability of the reals, describe a particular element of P(Words) that is not on this list. (B+S 3.4.13) Solution. We construct a set M that contains the element x if and only if the set paired with x does not contain the element x . So, for example, the set paired with “all” contains “all”, so M does not contain “all”. The set paired with “you” contains “you”, so M does not contain “you”. Continuing in this vein gives the set { you, found, it } . (I should have said “using the idea of Cantor’s proof of the power set theorem”; in a deep sense the uncountability of the reals and the power set theorem are the “same idea”, but this misled some people into trying to construct diagonal arguments.
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