If R and S are Two Equivalence Relations on a Set A, then R ∩ S is ……………………….
Question
If \( R \) and \( S \) are two equivalence relations on a set \( A \), then \( R \cap S \) is ……………………….
Solution
An equivalence relation must satisfy three properties:
- Reflexive
- Symmetric
- Transitive
Since both \( R \) and \( S \) are equivalence relations on \( A \), each of them satisfies all three properties.
Now consider the intersection:
\[ R \cap S \]
Any ordered pair belonging to \( R \cap S \) belongs to both \( R \) and \( S \).
Therefore:
- \( R \cap S \) is reflexive
- \( R \cap S \) is symmetric
- \( R \cap S \) is transitive
Hence, \( R \cap S \) is also an equivalence relation.
Therefore, the answer is:
\[ \boxed{\text{an equivalence relation}} \]