Exercise $4\mathrm{.}1\mathrm{.}1$: Prove that the following are not regular languages.

a$)\{0^$n$1^$nI$>=1\}.$This language, consisting of a string of O$\text{'}$s followed by an equal$-$length string of $1\text{'}$s$,$ is the language L$01$we considered informally at the beginning of the section. Here, you should apply the pumping lemma in the proof.

b$)$The set of strings of balanced parentheses. These are the strings of characters $"("$ and $")"$ that can appear in a well$-$formed arithmetic expression.

$*$ c$)\{0^$n$1^$nI$>=1\}.$ d$)\{0^$n$1^$m$2^$n I n and m are arbitrary integers$\}.$

e$)\{0^$n$1^$m I n$==1\}.$