Problem $1$: Let $i,j,k$ be positive integers. The alphabet for all the languages is $\{0,1\}.$

a$)$ Prove that the language that is comprised of all strings that have i number of $0\text{'}$s followed by j number of $1\text{'}$s and followed by k number of $0\text{'}$s$,$ where $k=i+j+5,$ is a not regular language using Pumping Lemma.

b$)$ Prove that the language that is comprised of all strings that have i number of a$\text{'}$s followed by $j$ number of b$\text{'}$s and followed by $k$ number of a$\text{'}$s$,$ where $i=j$ or $jk,$ is a not regular language using Pumping Lemma.

c$)$ Prove that the language that is comprised of all strings that have i number of a$\text{'}$s followed by j number of b $\text{'}$s$,$ where $ij,$ is a not regular language using Pumping Lemma.

d$)$ Prove that the language that is comprised of any string concatenated with itself $($e$.$g$.$ have the form ww$,$ where w is any string from $($:$\{a,b{\}}^{**}$