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

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.

$(a, b) \in R \cup S$ and $(b, c) \in R \cup S$

But $(a, c) \notin R \cup S$

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