$($a$)$ Give context$-$free grammars that generate the languages $L_1=\{a^i{b}^j{c}^k|i,j,k0,i=j\}$ and

$L_2=\{a^i{b}^j{c}^k|i,j,k0,j=k\}.$

$($b$)$ Show that $L_1{L}_2=\{a^n{b}^n{c}^n|n0\}.$

$($c$)$ Find the above language in the textbook, after the proof of the pumping Lemma $($write down

edition, page, number$).$ Show that the class of context$-$free languages is not closed under inter$-$

section.

$($d$)$ Use $($c$)$ and DeMorgan's law $($write down edition, page, number$)$ to prove that the class of

context$-$free languages is not closed under complementation.