Use the pumping lemma to demonstrate that L$1,$ L$2,$ L$3,$ L$4$ and L$5$ is not regular.

$($a$)$ L$1=\{$w in $\{0,1\}$

$$

: $0$

i$1$

j where i $<=$ j$\}(5$ points$)$

$($b$)$ L$2=\{$w in $\{$a$,$ b$,$ c$\}$

$$

: a

ib

jc

k$+2$ where i $=$ k and i$,$ j$,$ k $>=0\}(5$ points$)$

$($c$)$ L$3=\{$w in $\{0,1\}$

$$

: w is a palindrome.$\}(5$ points$)$

$($d$)$ L$4=\{$w$1$#w$2$ such that $|$w$1|=2|$w$2|,$ where $\backslash$Sigma $=\{0,1\}\}(5$ points$)$

$($e$)$ L$5=\{$w in $\{$a$\}$

$$

: a

$2$

n where n $>=0\}(5$ points$)$