1. Define a relation R on × ( \{0}) by (a, b) R (x, y) iff a y = bx. (a) Prove that R is an equivalence relation. (b) Describe the equivalence classes corresponding to R. 2. Let S = {1, 2, 3} and let...

1. Define a relation R on

× (
\{0}) by (a, b) R (x, y) iff a y = bx.

(a) Prove that R is an equivalence relation.

(b) Describe the equivalence classes corresponding to R.

2. Let S = {1, 2, 3} and let R be an equivalence relation on S.

(a) Describe the R with the fewest members. How many equivalence classes are there for this R ? Describe the corresponding partition of S.

(b) Describe the R with the fewest members such that (1, 2) is in R. How many equivalence classes are there for this R ? Describe the corresponding partition of S.

(c) Describe the R with the fewest members such that (1, 2) and (2, 3) are in R. How many equivalence classes are there for this R ? Describe the corresponding partition of S