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To determine if the given relation \( R = \{(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)\} \) on the set \( A = \{1, 2, 3\} \) is an equivalence relation, we need to check the following three properties:

1. **Reflexivity**: A relation \( R \) is reflexive if for every element \( a \in A \), the pair \( (a, a) \in R \).
   - Here, \( A = \{1, 2, 3\} \), and we have the pairs \( (1, 1), (2, 2), (3, 3) \) in \( R \), so the relation is reflexive.

2. **Symmetry**: A relation \( R \) is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \) as well.
   - The pairs \( (1, 2) \) and \( (2, 1) \) are both present in \( R \), and similarly, \( (2, 3) \) and \( (3, 2) \) are also present. Hence, the relation is symmetric.

3. **Transitivity**: A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \) as well.
   - Let's check:
     - For \( (1, 2) \in R \) and \( (2, 3) \in R \), we should have \( (1, 3) \in R \), but it is missing.
     - Therefore, the relation is **not transitive**.

Since the relation fails the transitivity property, it is **not an equivalence relation**.

Would you like more details or have any questions?

Here are five related questions:
1. What would make this relation transitive?
2. How do you determine if a relation is reflexive on any set?
3. Can a relation be symmetric but not transitive?
4. What are some examples of equivalence relations in everyday life?
5. How is an equivalence relation different from a partial order?

**Tip:** Reflexivity and symmetry are often easier to verify, but transitivity usually requires checking multiple pairs, making it the most intricate property of an equivalence relation.

Is the relation R = {(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)} on set A = {1,2,3} an equivalence relation?

Check if the relation R = {(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)} on set A = {1, 2, 3} satisfies the conditions for an equivalence relation by examining reflexivity, symmetry, and transitivity.

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