To check if they are equivalence relations, you need to check:

a. Every number is related to itself

b. If a is related to b, then b is related to a.

c. If a is related to b, and b is related to c, then a is related to c.

For R and S, these are fairly easy to check.

R∩S = {(1, 1), (2, 2), (3, 3)} which is an equivalence relation

R^(-1)∩S^(-1)

R^(-1) = {(1, 1), (2, 2), (2, 1), (1, 2), (3, 3)}

S^(-1) = {(1, 1), (2, 2), (3, 2), (2, 3), (3, 3)}

Which should look familar.

So that means we are left with D. Determine why D can't be an equivalence relation.

]]>(3, 3)} are two relations in the set X = {1, 2, 3}, the incorrect statement is:

(A) R and S are both equivalence relations

(B) R∩S is an equivalence relations

(C)R^(-1)∩S^(-1) is an equivalence relations

(D) R∪ S is an equivalence relations

With Regarks,

Prakash Panneer