Show that f is a symmetric relation on A

Question

I am learning about relations and I come across this exercise. And I don't understand the problem. Let me first state the problem here:

Let $f: A \rightarrow A$ be a function for which $f(f(x))=x$ for all $x\in A$. Show that $f$ is a symmetric relation on $A$.

First, I know that, to show that $f$ is a symmetric relation on $A$. I want to show that: for all $a,b \in A$ $aRb \rightarrow bRa$. Then, I will do:

Given that $aRb$, I fix arbitrary $a,b \in A$, then, I will know that $f(f(a))=a$. Then I am lost... I must have did something wrong. I would appreciate if you can give me a hint because I've stuck on this for a while.

Many thanks!

Answer 1

$aRb$ in your case means $f(a) = b$.

Now apply $f$ on both sides and you will get $a=f(f(a)) = f(b)$

Thus it holds $f(b)=a$ which is $bRa$