Question

prove right or wrong

“if r and s are transitive, then r n s is transitive”

Answer 1

Suppose (a, b) ∈ R∩S and (b, c) ∈ R∩S. We need to show that (a, c) ∈ R∩S. Because (a, b) ∈ R∩S, we know and (a, b) ∈ R and (a, b) ∈ S. Because (b, c) ∈ R∩S, we know and (b, c) ∈ R and (b, c) ∈ S. Since (a, b) ∈ R and (b, c) ∈ R and R is transitive we know that (a, c) ∈ R. Since (a, b) ∈ S and (b, c) ∈ S and S is transitive we know that (a, c) ∈ S. Since (a, c) ∈ R and (a, c) ∈ S, (a, c) ∈ R∩S. We have shown that whenever (a, b) ∈ R∩S, and (b, c) ∈ R∩S, (a, c) ∈R∩S , so R∩S is transitive