
Exercises
11. On the set ℝ consider the relation defined as “xRy if and only if x y is an integer.” Prove first that R
is an equivalence relation. Then find the equivalence classes of 1, −1.32, and π.
12. On the set A = {2, 3, 4, 5, 6,…} consider the rel ation defined as “xRy if and only if GCD(x, y) > 1.”
Is this an equivalence relation (explain how you reach your conclusion)? If it is, find [2], [3], and [8]
otherwise find some elements that are in relati on with 2, some in relation w ith 3, and some in rela-
tion with 8.
13. On the set A = {2, 3, 4, 5, 6,…} consider the relation defined as “xSy if and only if x and y have the
same prime factors. ...