Question

Let $A=\{1,2,3\}$ and $R=\{(1,2),(1,1),(2,3)\}$ be a relation on $A$. What minimum number of ordered pairs may be added to $R$, so that it may become a transitive relation on $A$?

Answer 1

Given, $A=\{1,2,3\}$ and $R=\{(1,2),(1,1),(2,3)\}$.

Now this relation is not transitive as $(1,2)\in R,(2,3)\in R$ but $(1,3)\notin R$.

So to make $R$ transitive we are to add this order pair.

Then the relation will be $R=\left\{\right(1,2),(1,1),(2,3),(1,3\left)\right\}$.

This relation $R$ is transitive.

So minimum one pair is to be added to make $R$ symmetric.