Proposition 2.12. Suppose- that R is an equivalence relation on a set A, and let a, b e A. Then the following are equivalent. (i) a ~b (i) a E (b) (iii) [a] n (b] + 0 (iv) [a] = [b]

(d) Give an example of an equivalence relation on the set {1,2, 3} with exactly two equivalence classes. (e) Given S is the set of integers (2, 3, 4, 6, 7, 9.. Let R be a relation defined on S by the following condition such that, for all r, y e S, xRy if 3|(x + y) which means 3 divides (x – y). i. Draw the digraph of R. ii. Say with reason whether or not R is • reflexive; • symmetric; • anti-symmetric; • transitive. In the cases where the given property does not hold, provide a counter example to justify this. i. is Ra partial order? Explain your answer. iv. is R an equivalence relation? (f) Letf : A → B and g : B → C be functions. Prove that if g o f is one-to-one, then f is one-to-one.