Question

Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number of ordered pairs so that the enlarged relation is symmeteric, transitive and reflexive.

Answer 1

We have,

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

R can be a transitive only when the elements (1, 3) is added

R can be a reflexive only when the elements (1, 1), (2, 2), (3, 3) are added

R can be a symmetric only when the elements (2, 1), (3, 1) and (3, 2) are added

So, the required enlarged relation, R' = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} = A $\times$ A