Problem with Functions and relations

Question

Here's the question:

Let $A$ be a nonempty finite set and let $f:A\rightarrow A$ be a function. Suppose that $f\circ f = Id_A$.

1. Let $R$ be a relation on $A$ defined by $$ aRb \iff a=b \lor f(a)=b $$

Prove that $R$ is an equivalence relation on $A$.
Find also the equivalence class $[a]$ for any $a\in A$.

2. Denote by $A_f$ the set of fixed points of $f$, that is, $$A_f = \{a \in A: f(a)=a\}$$

Use (1), or otherwise, prove that $\vert A\vert, \vert A_f\vert$ have the same parity.

I have completed the first part of part (1), and I believe the answer to the second part is $[a] = \{a,f(a)\}$ (correct me if I'm wrong).

However, I seem to be stuck on part (2) as I don't really get where I'm meant to start.

Answer 1

Here's an answer for Part B.

Let $X_1=Af=\{a\in A\,|\,f(a)=a\}$, and let $X_2=A-X_1=\{a\in A\,|\,f(a)\ne a\}$.

Note that $A=X_1\cup X_2$ and $X_1\cap X_2=\emptyset$. Hence $|A|=|X_1|+|X_2|$.

$A$ and $X_1=Af$ have the same parity because $|X_2|$ is even.

This is because $X_2$ a disjoint union of distinct pairs $\{a,f(a)\}$.

Answer 2

a modest contribution : proof of transitivity of R ( required for R to be an equivalence relation)