GATE Overflow Test Series | Discrete Mathematics | Test 1 | Question: 17

Question

Consider the following two sentences

• $S_1:$ The intersection of two transitive relations is transitive.
• $S_2:$ The union of two transitive relations is transitive.

Which of the below options is/are CORRECT?

• A. Only $S_1$
• B. Only $S_2$
• C. Neither $S_1$ nor $S_2$
• D. Both $S_1$ and $S_2$

Answer 1

Transitive relations are closed under intersection but not under union. Consider $R_1 = \{(1,2)\}$ and $R_2 = \{(2,3)\}.$ Now $R_1 \cup R_2 = \{(1,2), (2,3)\}$ which is not transitive as $(1,3)$ is not present despite $(1,2)$ and $(2,3)$ being there.