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$$ \delta(q,a) = \begin{cases} \delta_1(q,a) & q \notin F_1 \text{ or } a \neq \lambda \\ \delta_1(q,a) \cup \{q_1\} & q \in F_1 \text{ and } a = \lambda. \end{cases} $$

Look at the second line of the function definition. It's a transition function. $q_1$ is the starting state. What does that union mean here? Union of the function range with $\{q_1\}$?

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The function $\delta_1$ is the transition function of an NFA. It outputs a set of states. So $\delta_1(q,a) \cup \{q_1\}$ is just the union of two sets of states.

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