Math 310: Problem set 1

Instructions: This problem set is due Thursday, August 31, 2006. Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English.
  1. Read Chapter 1 of the text.
  2. Compute 350 mod 14. Show how to obtain your answer.
  3. Compute 350 mod 12. Show how to obtain your answer.
  4. Give an example of a relation R on a set S (you pick the R and S) that is reflexive and symmetric but not transitive. Justify your answers (i.e., define your S and R explicitly, explain why R is reflexive and symmetric, and give a specific example of elements for which transitivity fails).
  5. Give an example of a relation R on a set S that is reflexive and transitive but not symmetric. Justify your answers.
  6. Give an example of a relation R on a set S (you pick the R and S) that is transitive and symmetric but not reflexive. Justify your answers.
  7. Consider the relation R = {(a, b) : a and b are reals and 0 < |a - b| < 1} on the set S = R of real numbers. For each of the three properties of an equivalence relation, either show the property holds, or given an example for which it does not hold.
  8. Consider the relation R = {(a, b) : a and b are reals and |a - b| < 1} on the set S = R of real numbers. Is R an equivalence relation? Justify your answer.
  9. Consider the relation R = {(a, b) : a and b are integers and |a - b| < 1} on the set S = Z of integers. Is R an equivalence relation? Justify your answer.