hw5-solutions

If you object that “the smallest number contained

This preview shows pages 2–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: If you object that “the smallest number contained in T ” isn’t really a description of 2 that number in less than fifty characters, because we’d have to define T , then we could put a definition of T in the string. Unfortunately it’s hard to describe T in less than fifty characters! So we can describe T 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...
View Full Document

{[ snackBarMessage ]}

Page2 / 3

If you object that “the smallest number contained in T...

This preview shows document pages 2 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online