Problem $2$: Let $i,j,k$ be positive integers. The alphabet for all the languages is $\{a,b\},$ unless otherwise stated. For each language listed below provide a CFG $($you can generate the CFG from scratch, or generate a CFG as a combination of other CFG$\text{'}$s$)$

a$)$ 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 $ji+1.$

b$)$ 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 $j2$i and $j3i.$

c$)($The alphabet for this language is $\{a,b,c\})$ 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 c$\text{'}$s$,$ where $k=i+j.$

d$)$ The language is comprised of all string of the form wuvur ${?}^R,$ where $w$ and $u$ are from $\{a,b{\}}^{+}$and the $|w|=|v|=2.$