Determine if language $L=\{xyxz\,\,|\,\,x\neq\varepsilon,\,\,x,y,z\in\{0,1\}^{*}\}$ is regular

Question

Question.

Determine if the language is regular:

$L=\{xyxz\,\,|\,\,x\neq\varepsilon,\,\,x,y,z\in\{0,1\}^{*}\}$

I think $L$ is non regular, because of the second x.

I'm trying to prove with the Pumping Lemma, but I'm not so sure how to choose the word.

Answer 1

Hint. You already observed that $$ \varepsilon \cup 01^* \cup 10^* \subseteq L^c $$ Now use the fact that $$ 0\{0,1\}^*0\{0,1\}^* \cup 1\{0,1\}^*1\{0,1\}^* \subseteq L $$ to conclude.