For a set $S\subseteq Q_E$, let $E(S)$ be its <epsilon closure>, and put
$$
\delta_E(S,a)=\bigcup_{q\in S}\delta_E(q,a).
$$
The <subset construction with epsilon transitions> gives the <deterministic finite automaton>
$$
Q_D=\mathcal P(Q_E),
\qquad q_D=E(\{q_0\}),
$$
$$
\boxed{\delta_D(S,a)=E\bigl(\delta_E(S,a)\bigr),
\qquad
F_D=\{S\subseteq Q_E:S\cap F_E\ne\varnothing\}.}
$$
Unreachable subsets may be deleted from $Q_D$ without changing the recognized language.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 4G / a / Solution

The subset construction converts an epsilon-NFA into a <deterministic finite automaton>. Its states are subsets of NFA states, its initial state is the <epsilon closure> of the NFA initial state, and every symbol transition is followed by another epsilon closure.

Subset construction with epsilon transitions

{title2=$E(S)$}

The epsilon closure of a state set $S$ consists of every state reachable from $S$ using zero or more epsilon transitions.


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 Incoming links: Epsilon closure