For X = {1, 2, 3, 4, 5},?

Question

Determine if each of the following relations is an equivalence relation. If
yes, explain your answer and give equivalence classes. If no, explain why.


(i) R1 = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3) (3, 1), (3, 2), (3, 3), (4, 4), (5, 5) }




(ii) R2 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (3, 1), (3, 3), (4, 1), (4, 4), (5, 5)}

Answer 1

(i)Yes. To see this, we have to verify that Reflexivity, Symmetry and Transitivity hold for R1.

R1 is reflexive since (1,1), (2, 2), (3, 3), (4, 4), (5, 5) belong to R1

R1 is symmetric since [ (1,2), (2, 1)], [(1, 3), (3, 1)], [(2, 3), (3,2)], [(3, 1), (1, 3)] are members of R1

R1 is transitive since for any two pairs s.t (a, b) & (b, c), we also have (a, c). For instance (1, 2), (2, 3) imply that 3 is equivalent to 1. Indeed, we have (3, 1).

The equivalence sets for R1 are {1, 2, 3} , {4}, and {5}

(ii) No. The collection {(1, 2), (1, 3)} implies that if R2 were an equivalence relation, we would have had the pair (2, 3) in R2. But (2, 3) is not present. Hence R2 cannot be an equivalence relation.

Answer 2

A relation is an equivalence relation if it is reflexive, symmetric and transitive. Test all 3 of these things on each relation. If the relation satisfies all 3 of these properties then it is an equivalence relation.

Hint: one of them is an equivalence relation and the other one is not.