Question

If R and S are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.

Answer 1

Let A = {a, b, c} and R and S be two relations on A, given by

R = {(a, a), (a, b), (b, a), (b, b)} and
S = {(b, b), (b, c), (c, b), (c, c)}

Here, the relations R and S are transitive on A.

$$\begin{gathered} \left(a,b\right)\in R\cup S\mathrm{and}\left(b,c\right)\in R\cup S \\ \mathrm{But}\left(a,c\right)\notin R\cup S \end{gathered}$$

Hence, R$\cup$S is not a transitive relation on A.