Question

Given a relation $R=(1,2),(2,3)$ on the set of natural numbers, add a minimum number of ordered pairs so that the enlarged relation is symmetric, transitive and reflexive.

Answer 1

For the relation R to be reflexive, it is necessary that (n, n) $\in$ R for every n $\in$ N that is, R must have all pairs (1, 1), (2, 2), (3, 3),...........

For R to be symmetric, it must contain the pair (2, 1) and (3, 2) since the pairs (1, 2) and (2, 3) are already present.

For R to be transitive, it must contain the pair (1, 3) since (1, 2) and (2, 3) are already there. We must then also include the pair (3, 1) for symmetry. Hence the relation R' obtained from R by adding a minimum number of ordered pairs to R to make it an equivalence relation is

$R^{\prime }=(1,2).(2,1),(2,3),(3,2),(1,3),(3,1),(1,1),(2,2),(3,3),.........$