Chapter $8$:

Show that the following languages on $=\{a,b,c\}$ are not context$-$free. Use Pumping

Lemma Conditions:

$($a$)L=\{a^n{b}^j$:$n{j}^2\}.$

$($b$)L=\{a^n{b}^j$:$n(j-1{)}^3\}.$

$($c$)**L=\{a^n{b}^j{c}^k$:$k=jn\}.$

$($d$)L=\{a^n{b}^j{c}^k$:$k>n,k>j\}.$

$($e$)L=\{w$:$n_a(w)=n_b(w)**n_c(w)\}L=\{win\{a,b,c{\}}^{**}$:$n_a(w)+n_b(w)=2n_c(w),n_a(w)=n_b(w)\}****L=\{\begin{array}{c}{a}^n{b}^m\text{:}n\end{array}m$