GATE CS 2017 Set 2 Q16 — Context-free Grammars and Push-down Automata

To find the language generated by the given grammar, let us analyze the languages generated by the non-terminals $X$ and $Y$ individually:

1. Language generated by $X$:

   The production rules for $X$ are:

   $$X \rightarrow aX \mid a$$

   This generates one or more $a$'s. Thus,

   $$L(X) = \{a^k \mid k \geq 1\}$$
2. Language generated by $Y$:

   The production rules for $Y$ are:

   $$Y \rightarrow aYb \mid \epsilon$$

   This generates equal numbers of $a$'s followed by $b$'s, including the empty string. Thus,

   $$L(Y) = \{a^n b^n \mid n \geq 0\}$$

Since the start symbol $S$ has the production $S \rightarrow XY$, the language $L(S)$ is the concatenation of $L(X)$ and $L(Y)$:

Let $m = k + n$. Since $k \geq 1$, we have:

Since $n \geq 0$, the language can be written as:

Therefore, the correct option is C.