Question

Let R and S be two non-void relations on a set A. Which of the following statements is false

• A. R and S are transitive then R $\cup$ S is not transitive
• B. R and S are transitive then R $\cap$ S is transitive
• C. R and S are symmetric then R $\cup$ S is symmetric
• D. R and S are reflexive then R $\cup$ S is reflexive

Answer 1

The correct option is A

R and S are transitive then R $\cup$ S is not transitive

let A = {1,2,3} and R = {(1.1).(1,2)}. S = {(2. 2) (2, 3)} be transitive relations on A.

Then R $\cup$ S = {(1, I); (1, 2); (2.. 2); (2. 3)}

Obviously, R $\cup$ S is not transitive. Since (1, 2) $\in$ R $\cup$ S and

(2,3) $\in$ R $\cup$ S but (1, 3) $\in$ R $\cup$ S.