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 $$aRb \land bRa$$ $$bRc \land cRb$$
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!