Math Homework #2

This homework assignment is due Wednesday, October 5^th by 4 p.m., either to me in my office or else in our Homework Box. You may speak to your classmates and others (and me J) about these problems, but your write-up should be your own. Show your work!

Do the following problems by hand, from Schaum, pp. 301-303:

32, 33, 36, 40, 43, 47, 52, 58, 61, 68

  Consider the real number 3.5656565656... . Express this number as a ratio of integers, using one of the two methods we used in class.

 The following problem is a variation on the Cantor diagonalization argument we did in class. You are to argue that the set of all subsets of the set of positive integers N is not countable. Assume BWOC that it is countable, and so these sets can be listed in an enumeration S₁, S₂, S₃, S₄, … . Now build a subset of N not in the list, by ensuring at each step that the subset S_n is not equal to the new set. How? Put n in if it is out, or out if it is in!!! (This last statement is intentionally weird --- think about it).