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 ⇒ R∪S is transitive
• B. R and S are transitive ⇒ R∩S is transitive
• C. R and S are symmetric ⇒ R∪S is symmetric
• D. R and S are reflexive ⇒ R∩S is reflexive

Answer 1

Answer: (A)

R and S are transitive ⇒ R∪S is transitive

Answer 2

Let $A = \{ 1,2,3 \}$ and R = {(1, 1), (1, 2)},

S = {(2, 2) (2, 3)} be transitive relations on A.

Then R ∪ S = {(1, 1); (1, 2); (2, 2); (2, 3)}

Obviously, R ∪ S is not transitive. Since (1, 2) $\in$ R ∪ S and $( 2,3 ) \in R \cup S$ but (1, 3) $\notin R \cup S$.