hw5-solutions

# As say the smallest number which cannot be described

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as, say, “the smallest number which cannot be described in one hundred characters” (which contains 71 characters) and replace 50 with 100; the paradox remains intact. 5. Suppose we used our failed perfect shuffling of digits to mix the digits of the numbers ( x, y ) that describe a point on the square to get a (almost) one-to-one correspondence with the points on a line segment. (a) What point on the square would be paired with the point 0 . 120001000100010001 . . . ? With which point on the line does that point actually get paired? Is this a problem? Explain. (b) Repeat (a) for 0 . 12001001001001 . . . . Solution. (a) Reading off alternate digits from the number above, we get (0 . 100000 . . . , . 2010101 . . . ). But in setting up our one-to-one correspondence we declared that in case of ambiguity, we will always use the expansion of a number which ends in infinitely many nines. So this point in the square actually gets paired by treating it as (0 . 099999 . . . , . 2010101 . . . ) and so is paired with 0 . 0290919091 . . . . Thus this “one-to-one correspondence” fails to actually be one! (b) Reading off alternate digits gives (0 . 101001001 . . . , . 2001001001 . . . ), which is not a problem since neither of these ends in infinitely many zeroes. (B+S 3.5.11-12) 6. Prove that the cardinalities of points in the following two geometrical objects are equal (these objects are made up of little line segments – so they have no thickness) T L Solution. Each letter consists of a horizontal segment and a vertical segment. So we can construct a one-to-one correspondence between L and T by pairing the points in the horizon- tal segments with each other and the points in the vertical segments with each other. (B+S 3.5.17) 3
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