To determine whether the given relation $R = \{(1,1), (2,2), (3,3), (4,4), (5,5), (1,5), (5,1), (3,5), (5,3), (1,3), (3,1)\}$ on the set $\{1, 2, 3, 4, 5\}$ is an equivalence relation, we need to check if it satisfies the three necessary properties of an equivalence relation:

1. Reflexivity: A relation $R$ is reflexive if for every element $a \in A$, the pair $(a, a)$ is in $R$.
2. Symmetry: A relation $R$ is symmetric if for every pair $(a, b) \in R$, the pair $(b, a)$ is also in $R$.
3. Transitivity: A relation $R$ is transitive if whenever $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.

1. Reflexivity:

For the set $\{1, 2, 3, 4, 5\}$, we need to ensure that each element is related to itself. From the given relation, we have the pairs $(1,1), (2,2), (3,3), (4,4), (5,5)$, which satisfy reflexivity.

2. Symmetry:

The given relation contains the pairs:

• $(1,5)$ and $(5,1)$
• $(3,5)$ and $(5,3)$
• $(1,3)$ and $(3,1)$

Since for every pair $(a, b) \in R$, we also have $(b, a) \in R$, the relation is symmetric.

3. Transitivity:

We need to verify whether the relation satisfies transitivity. That is, if $(a, b) \in R$ and $(b, c) \in R$, then $(a, c)$ should also be in $R$.

• $(1, 5)$ and $(5, 3)$ imply $(1, 3)$, which is in $R$.
• $(3, 5)$ and $(5, 1)$ imply $(3, 1)$, which is in $R$.
• $(1, 3)$ and $(3, 5)$ imply $(1, 5)$, which is in $R$.
• All other possible combinations of pairs from the relation are also satisfied.

Thus, the relation is transitive.

Conclusion:

Since the relation is reflexive, symmetric, and transitive, it is an equivalence relation.

Equivalence Classes:

The equivalence classes are formed by grouping all elements that are related to each other. From the given pairs in the relation, we have the following equivalence classes:

• $\{1, 3, 5\}$ (since $1$, $3$, and $5$ are related to each other)
• $\{2\}$ (since $2$ is only related to itself)
• $\{4\}$ (since $4$ is only related to itself)

Thus, the equivalence classes are:

• $\{1, 3, 5\}$
• $\{2\}$
• $\{4\}$

Let me know if you'd like further details or have any questions!

Further Related Questions:

1. How do you prove that a relation is symmetric?
2. What is the significance of an equivalence relation in set theory?
3. Can a relation be reflexive and symmetric but not transitive?
4. How do you compute the equivalence classes for a larger set or relation?
5. What are the applications of equivalence relations in mathematics?

Tip: Always check the reflexive property first when determining if a relation is an equivalence relation, as it’s the easiest to verify for all elements in the set.