If relation $R^2$ is symmetric, R does not need to be symmetric

Question

I was checking a question that said the following

If $R^2$ is symmetric, does that mean that R is symmetric?

This being R a relation on the set $A$.

What I thought the proof would be is:

$$aR^2c \land cR^2a$$ This because $R^2$ is symmetric.

From the definition of $R^2$:

$$aRb \land bRc \iff aR^2c$$ $$bRa \land cRb \iff cR^2a$$

Here we can see that R is symmetric $$aRb \land bRa$$ $$bRc \land cRb$$

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However... this is not true... it does not mean that R is symmetric (I can give an example).

What did I miss in my proof?

Thank you for your help!

Answer 1

$R^2$ being symmetric does not mean that $aR^2c\land cR^2a$ for arbitrary $a,c\in A$. It means that if $aR^2c$, then $cR^2a$. Equivalently, it means that if $a,c\in A$, then either $aR^2c\land cR^2a$, or $a\not R^2c\land c\not R^2a$.