3. Recall that an equivalence relation must be reflexive, symmetric and transitive. The solution for Section 10.2, Question 1 is summarized below:
R1 = { (1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (4,4) } is not reflexive, not symmetric and not transitive.
R2 = { (1,1), (1,2), (2,1), (2,2), (3,3), (4,4) } is reflexive, symmetric and transitive.
R3 = { (1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4)} is reflexive and symmetric but not transitive.
R4 = { (2,1), (3,1), (3,2), (4,1), (4,2), (4,3) } is not reflexive and not symmetric but is transitive.
R5 = { (1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4)} is reflexive and transitive but not symmetric.
R6 = { (1,1), (2,2), (3,3), (4,4) } is reflexive, symmetric and transitive.
Thus R2 and R6 are equivalence relations. Now draw the directed graphs for each of these relations. Then find the required paritions of the set {1, 2, 3, 4} and list the equivalence classes.