Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 4J / a

Let $\Sigma =\{a,b\}$, $V=\{S\}$, andThen $\mathcal{L}(G)=\{a^{n}b{a}^n:n\ge 0\}$. Adding $S\to \epsilon$ also produces $a^{2n}$; for example $aa$ is generated but lies in neither $\mathcal{L}(G)$ nor $\{\epsilon \}$. Thus the proposed equality can fail because the new erasing rule may be used on a noninitial occurrence of $S$.