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

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

Solution:

Let  = {abc} and R and S be two relations on A, given by

= {(aa), (ab), (ba), (bb)} and
= {(bb), (bc), (cb), (cc)}

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 $A$.