(a) Let $E=\left(Q_{E}, \Sigma, \delta_{E}, q_{0}, F_{E}\right)$ be a nondeterministic finite-state automaton with $\epsilon$-transitions $(\epsilon$-NFA). Define the deterministic finite-state automaton (DFA) $D=\left(Q_{D}, \Sigma, \delta_{D}, q_{D}, F_{D}\right)$ obtained from $E$ via the subset construction with $\epsilon$-transitions.

(b) Let $E$ and $D$ be as above. By inducting on lengths of words, prove that

$$\hat{\delta}_{E}\left(q_{0}, w\right)=\hat{\delta}_{D}\left(q_{D}, w\right) \text { for all } w \in \Sigma^{*}$$

(c) Deduce that $\mathcal{L}(D)=\mathcal{L}(E)$.