## 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.
- Read Chapter 1 of the text.
- Compute 3
^{50} mod 14. Show how to obtain your answer.
- Compute 3
^{50} mod 12. Show how to obtain your answer.
- 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).
- Give an example of a relation R on a set S
that is reflexive and transitive but not symmetric. Justify your answers.
- 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.
- 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.
- 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.
- 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.