A). There exists a bi-implication between the languages $L_1 = \{ a^mb^n\mid m,n\ge 1 \}$ and $L_2 = a^+b^+$.
But as such no kind of bi-implication exists between the languages in part B)
B). Are the two languages $L_1 = \{a^m(bc^k)^n \mid m,n \geq 1\}$ and $L_2 = a^+(bc^+)^+$ same ?
No, the languages are not same and in fact the first one is DCFL and not regular while the second one is a regular language.
Can anyone explain why is this happening ?