Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 4I / c / Solution

No. Context-free languages are not closed under symmetric difference (a standard counterexample is obtained from the two CFLs enforcing respectively equality of the first two and last two blocks in $a^{\ast }{b}^{\ast }{c}^{\ast }$). Transport such a pair through the bijection $\beta$; pointwise addition corresponds exactly to symmetric difference.