Let r,s,u be binary relations in U ...Verify following proreperty;

Question

Which explanation to understand this do i need to learn ?

" if both relations r and s are transitive then the intersection of r and s is transitive too."

Answer 1

$ (a,b), (b,c) \in R \cap S \implies \left\{ \begin{array}{l} (a,b), (b,c) \in R \overset{R \text{ transitive}}{\implies} (a,c) \in R \\ (a,b), (b,c) \in S \overset{S \text{ transitive}}{\implies} (a,c) \in S \\ \end{array} \right\} \implies (a,c) \in R \cap S $

Therefore, $R \cap S$ is also transitive.