We will prove this using an example.

Let A = {a, b, c} be a set and

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

S = {(a, a) (b, b) (c, c) (b, c) (c, d)} are two relations on A

Clearly R and S are transitive relation on A

Now,

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

Here, (a, b) ∈ R ⋃ S and (b, c) ∈ R ⋃ S

but (a, c) ∉ R ⋃ S

∴ R ⋃ S is not transitive