A) R and S are transitive \[\Rightarrow \text{ }R\text{ }\cup \text{ }S\] is transitive
B) R and S are transitive \[\Rightarrow \text{ }R\text{ }\cap \text{ }S\] is transitive
C) R and S are symmetric \[\Rightarrow \text{ }R\text{ }\cup \text{ }S\] is symmetric
D) R and S are reflexive \[\Rightarrow \text{ }R\text{ }\cap \text{ }S\] is reflexive
Correct Answer: A
Solution :
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) \[\notin R\cup S\].
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