The language $$\left\{ {{a^m}{b^n}{c^{m + n}}\left| {m,n \ge } \right.} \right\}$$ is

Let $$M = \left( {K,\,\sum {,\,F,\,\Delta ,\,s,\,F} } \right)$$ be a pushdown automation. Where $$K = \left\{ {s,\,f} \right\},\,F = \left\{ f \right\},\,\sum { = \left\{ {a,b} \right\},\,F = \left\{ a \right\}} $$ and $$\Delta = \left\{ {\left( {\left( {s,\,a,\, \in } \right)} \right.,\,\left. {\left( {s,\,a} \right)} \right),\,\left( {\left( {s,\,b,\, \in } \right),\,\left. {\left( {s,\,a} \right)} \right),\,} \right.} \right.$$ $$\left( {\left( {s,\,a,\, \in } \right),\,\left( {f,\, \in } \right),\,\left( {\left( {f,\,a,\,a} \right),\,\left. {\left( {f,\, \in } \right)} \right),\,\left( {\left( {f,\,b,\,a} \right),\,\left. {\left. {\left( {f,\, \in } \right)} \right)} \right\}} \right.} \right.} \right..$$

Which one of the following strings is not a number of $$L(M)?$$

Consider the following grammar $$G:$$
$$\eqalign{ & S \to bS\,\left| {\,aA\,\left| {\,b} \right.} \right. \cr & A \to bA\,\left| {\,aB} \right. \cr & B \to bB\,\left| {\,aS\,\left| {\,a} \right.} \right. \cr} $$

Let $${N_a}\left( w \right)$$ and $${N_b}\left( w \right)$$ denote the number of $$a's$$ and $$b's$$ in a string $$w$$ respectively. The language
$$L\left( G \right)\,\,\, \subseteq \left\{ {a,b} \right\} + $$ generated by $$G$$ is

Which of the following grammar rules violate the requirements of an operator grammar ? $$P,$$ $$Q, R$$ are non-terminals and $$r, s, t$$ are terminals. $$$\eqalign{ & 1)\,\,\,P \to Q\,R\,\,\,\,\,2)\,\,\,P \to Q\,s\,R \cr & 3)\,\,\,P \to c\,\,\,\,\,\,\,\,\,\,\,4)P \to Q\,t\,R\,r \cr} $$$