• # question_answer 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 $\Rightarrow \text{ }R\text{ }\cup \text{ }S$ is transitiveB) R and S are transitive $\Rightarrow \text{ }R\text{ }\cap \text{ }S$ is transitiveC) R and S are symmetric $\Rightarrow \text{ }R\text{ }\cup \text{ }S$ is symmetricD) R and S are reflexive $\Rightarrow \text{ }R\text{ }\cap \text{ }S$ is reflexive

Let $A=\{1,\,2,\,3\}$ and R = {(1, 1), (1, 2)}, S = {(2, 2) (2, 3)} be transitive relations on A. Then R U S = {(1, 1); (1, 2); (2, 2); (2, 3)} Obviously, R U S is not transitive. Since (1, 2) $\in$ R U S and $(2,\,3)\in R\cup S$ but (1, 3) $\notin R\cup S$.