I have two questions about how to use pumping lemma for regular languages to show that two languages are not regular. I would appreciate if someone can confirm if my answers make sense, and if not, what needs to be done
1) Prove that the following languages aren't regular. You can use the pumping lemma and the closing of the class of the regular languages under union, intersection and complement.
a) $L_1 = \{w \in \{0,1\}^* \mid w \text{ is not a palindrome}\}$
My solution: Pick up w as the string $010010$. $x = \varepsilon$, $y = 01$ and $z = 0010$. By the pumping lemma the string $xyyz$ should also be regular. However $xyyz = 01010010$ which is not a palindrome. So, by contradiction, $L_1$ is not a regular language.
b) $L_2 = \{ wtw \mid w,t \in \{0,1\}^* \}$
My solution. Pick up w as the string $0101$ and pick up $t=011$ as the other string. So $wtw= 0101\ 011\ 0101$ which belongs to $\{0,1\}$. By the pumping lemma, the string $xyyz$ should also be regular. However, choosing $y=110$ of $010\ 0|11\ 0|0101$, the string $0101\ 011\ 110\ 0101$ is produced which cannot be generated by $wtw$. So, by contradiction, $L_2$ is not a regular language.
Many thanks to any help.
