I've been trying to understand the idea behind proving a language is not regular by using Nerode's theorem, but I just couldn't apply the idea on what I've been asked.
The problem is to prove the following languages are non-regular:
$L_1 = \{waw^R \mid w∈{a,b}^* \}$, the language of all odd-length palindromes over $\{a,b\}$ such that in the middle of the palindrome there is an $a$.
$L_2=\{a^i b^j c^{i+j} \mid i,j∈\mathbb N\}$
$L_3=\{a^{2n} b^n \mid n∈\mathbb N\}$
$L_4=\{w^2 \mid w∈\{a,b\}^*, \text{$w$ is a palindrome}\}$
Any help will be appreciated.
