Question $5$: $(\frac{20}{100})$ : Below we will provide you with a number of defi$-$

nitions of languages. Your job is to apply the pumping lemma to prove that

the languages are not regular.

a$)$ contains an equal number of $0s$ and $1s,$ also described as

$L=\{0^n{1}^n|n0\}$

b$)L=\{0^n{1}^{2n}{0}^n|n0\}.$

c$)$ Consider the following languages over the alphabet $=\{a,b,c\}$ :

$A=\{\begin{array}{c}{a}^n{b}^m{c}^{nm}|n\\ m0\end{array}$

$B=\{a^{2n}{b}^m{c}^{k+1}{a}^2|n,m,k0\}$

One of those languages is regular. Find a regular expression $($RE$)$ for

it$.$ The other is not regular. Use the pumping lemma to prove that.

Justify all your answers and show all your work.