$$\forall \sigma,\beta \in S_5 \quad \sigma\,R\, \beta \Leftrightarrow \quad \sigma(1)=\beta(1) \land \sigma(5)=\beta(5) $$
1) Prove that $R$ is an equivalence relation: The reflexivity,symmetry and transitivity of $R$ coming from the reflexivity, symmetry and transitivity of $ \quad = \quad$ right?
2) Find the equivalence class of $[i_{S_5}]_R$ where $i_{S_5}$ is the identity permutation.
I have no clue how to solve the point 2). Can anyone could explain me the process? thanks
$(1)$ Correct; it amounts to recognizing the relation is one of equality, which we know is reflexive, symmetric, and transitive.
$(2)$ $$ \forall \sigma,\beta \in S_5: \quad \sigma \,R\,\beta \iff \left(\sigma(1)=\beta(1)) \land (\sigma(5)=\beta(5)\right) $$
Now, we are looking for all permutations $\sigma\in S_5$ such that:
$id(1) = 1 = \sigma(1),$ and also
$id(5) = 5 = \sigma(5)$
In other words, we are looking for all permutations that "fix" both $1$ and $5$: All permutations that permute 1 to 1, and 5 to 5.
$\left [id_{s_5}\right]$ includes $(2,3), (2,4), (3, 4), (2, 3, 4)$.
More formerly, the equivalence class $\left[id_{s_5}\right] = \left\{id_{s_5}, (2, 3), (2, 4), (3, 4), (2,3,4)\right\}$