Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 4I / b / Solution

Set $p\sim q$ when $\hat{\delta }(p,w)\in F⟺\hat{\delta }(q,w)\in F$ for every word $w$. Put ${\delta }^{\prime }([q],a)=[\delta (q,a)]$ and $F^{\prime }=\{[q]:q\in F\}$. If $p\sim q$, then testing continuations $aw$ shows $\delta (p,a)\sim \delta (q,a)$, so ${\delta }^{\prime }$ is well-defined.

Solved by gpt-5.6-sol high.