Question

A={1,2,3,4} and R={(1,2),(3,4),(2,1)}.find transitive closure of R

Answer 1

Let $A=\{1,2,3,4\}$ and $R=\{(1,2),(3,4),(2,1)\}$. Let us find transitive closure of $R.$ Since $(1,2)\in R$ and $(2,1)\in R$, we conclude that $(1,1)$ must belong to the transitive closure of $R.$ By analogy, $(2,1)\in R$ and $(1,2)\in R$ imply $(2,2)$ must belong to the transitive closure of $R.$ It is easy to see that the relation $\{(1,2),(3,4),(2,1),(1,1),(2,2)\}$ is transitive, and hence it is the transitive closure of $R.$