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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.
Solution
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) ∈ R ∪ S and (b, c) ∈ R ∪ S
But (a, c) ∉ R ∪ S
Hence, R ∪ S is not a transitive relation on A.
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