Quiz 5

Show all work in a neat and organized fashion. Clearly indicate your answers.

10 points possible.

1. (4 pts.) Let n be a positive integer. Prove that congruence modulo n is a symmetric relation on the set of integers.

2. (3 pts.) Prove that if w and b are both odd integers, then w º b mod 2.

3. (3 pts.) To draw a picture of a relation R on a set A, we make a diagram in which each element of A is represented by a dot. If a R b, then we draw an arrow from dot a to dot b. If it should happen that b is also related to a, we draw another arrow from b to a. And if a R a, then we draw a looping arrow from a to itself.

Let A={1,2,3,4,5}, and let R be the relation ³ . Draw a picture of this relation.

Optional Bonus Problem. (3 optional bonus points possible.) In terms of the picture (as in Problem 3) of a relation on a set, what does it mean for a relation to be reflexive? symmetric? transitive?