GATE Overflow Test Series | Discrete Mathematics | Test 1 | Question: 18

Question

If $R$ and $S$ are two equivalence relations on a set $A,$ then which of the below statements is/are TRUE?

• $S_1: R \cup S$ is an equivalence relation.
• $S_2: R \cap S$ is an equivalence relation.

• A. $S_1$ is true and $S_2$ is false
• B. $S_1$ is false and $S_2$ is true
• C. Neither $S_1$ nor $S_2$ is true
• D. Both $S_1$ and $S_2$ are true

Answer 1

Statement $S_1$ is false as Transitivity may be lost due to union.

For $R \cup S:$

1. Reflexive : true
2. Transitive: false
3. Symmetric: true

For $R \cap S:$

1. Reflexive : true
2. Transitive: true
3. Symmetric: true

Statement $S_2$ is true .