Question

Given a relation $R=\{(1,2),(2,3)\}$ on the set of natural numbers,Find the minimum number of ordered pairs should be added so that the enlarged relation is symmetric,transitive and reflexive.

Answer 1

For the relation R to be reflexive, it is necessary that $(n,n)ϵR$ for every $nϵN$ 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 $R^{\prime }$ obtained from R by adding a minimum number of ordered pairs to R to make it an equivalence relation is
$R^{\prime }=\{(1,2),(2,1),(2,3),(3,2),(1,3),(3,1),(1,1),(2,2),(3,3)\}$