$L = \{ xwyw^r \mid x,w,y \in (a+b)^+ \}$ where $w^r$ is the reverse string of $w$.
If we take $w = {}$minimum string possible ${} = a$ or $b,$ I think it could be regular
Lets say $w=a$, then RE could be $(a+b)^+a(a+b)^+a$
And for $w=b$, then RE could be $(a+b)^+b(a+b)^+b$
So Overall: $(a+b)^+a(a+b)^+a + (a+b)^+b(a+b)^+b$
Is this enough proof for being regular? While trying to prove non-regularity,
I am not able to prove this language as non-regular.
Any thoughts on this? Thank you.
You are quite right. But you'll probably have to provide a proof.
A frequent proof technique to show that $L=L'$ is to prove that both $L\subset L'$ and $L'\subset L$. In other words,$\forall \omega.\omega \in L\implies\omega\in L' \land \omega\in L'\implies\omega\in L$.
It's evident that every string in your language is in $L$. That leaves you to prove that every string in $L$ is in your language. You can most easily do this by rewriting $w$ as $w_1w'$ where $w'\in \{a,b\}^*$ and $w_1$ is the first symbol in $w$. Then $$xwyw^R = xw_1w'yw'^Rw_1$$ which is in your language because $w'yw'^R \in \{a, b\}^+$.