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

a$)$ Prove that the language that is comprised of all strings that have i number of $0\text{'}$s followed by $\backslash ($ j $\backslash )$ number of $1\text{'}$s and followed by $\backslash ($ k $\backslash )$ number of $0\text{'}$s$,$ where $\backslash ($ k$=$i$+$j$+5\backslash ),$ 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 $\backslash ($ j $\backslash )$ number of b$\text{'}$s and followed by $\backslash ($ k $\backslash )$ number of a$\text{'}$s$,$ where $\backslash ($ i$=$j $\backslash )$ or $\backslash ($ j

eq k $\backslash ),$ 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 $\backslash ($ j $\backslash )$ number of b$\text{'}$s$,$ where $\backslash ($ i $\backslash$leq j $\backslash ),$ 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 $\backslash (\backslash$left$.\backslash \{\backslash$mathrm$\{$a$\},\backslash$mathrm$\{$b$\}\backslash \}^\{*\}\backslash$right$)\backslash )$ is a not regular language using Pumping Lemma.