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.