Math 310: Problem set 8

Instructions: This problem set is due Thursday, November 9, 2006. Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English.
  1. Let R be the ring whose elements are continuous functions f:[0,1] -> R (i.e., real valued functions y=f(x) defined on the interval 0 <= x <= 1. Thus x has to be between 0 and 1 inclusive, and for each such x, the value f(x) of f at x has to be real, and f:[0,1] -> R has to be continuous).
  2. Let b, c and d be positive integers bigger than 1. Suppose we try to define a map h:Z/bZ -> Z/cZ x Z/dZ by specifying that h([x]b) = ([x]c,[x]d).
  3. Let b = 210, c = 14 and d = 15. Let h:Z/bZ -> Z/cZ x Z/dZ be defined as in the previous problem. Find an integer n such that h([n]b) = ([5]c,[12]d). Explain how you obtain your answer.