An equivalence relation R in A divides it into equivalence classes` A_1 ,A_2 ,A_3`What is the value of`A_1 uu A_2 uu A_3 and A_1 nn A_2 nn A_3`

Answer

To solve the question regarding the equivalence classes \( A_1, A_2, \) and \( A_3 \) formed by the equivalence relation \( R \) in set \( A \), we will analyze the properties of equivalence classes and their unions and intersections.

### Step-by-Step Solution:

1. **Understanding Equivalence Classes**:
- An equivalence relation on a set \( A \) partitions \( A \) into disjoint subsets known as equivalence classes. Each element of \( A \) belongs to exactly one equivalence class.

2. **Identify the Union of Equivalence Classes**:
- The union of the equivalence classes \( A_1, A_2, \) and \( A_3 \) is represented as:
\[ A_1 \cup A_2 \cup A_3 \]
- Since these classes are disjoint and cover the entire set \( A \), the union of all equivalence classes will equal the original set \( A \):
\[ A_1 \cup A_2 \cup A_3 = A \]

3. **Identify the Intersection of Equivalence Classes**:
- The intersection of the equivalence classes \( A_1, A_2, \) and \( A_3 \) is represented as:
\[ A_1 \cap A_2 \cap A_3 \]
- Since the equivalence classes are disjoint, they do not share any elements. Therefore, the intersection of any two or more disjoint sets is the empty set:
\[ A_1 \cap A_2 \cap A_3 = \emptyset \]

### Final Answers:
- The value of \( A_1 \cup A_2 \cup A_3 \) is \( A \).
- The value of \( A_1 \cap A_2 \cap A_3 \) is \( \emptyset \).