$1\mathrm{.}4.$ For each automaton A$,$ let LA denote the language of all the words accepted by this automaton, i$.$e$.,$ of all the words for which this automaton ends up in a final state. In class, we learned a general algorithm that:

given two automata A and B that correspond to languages LA and LB$,$

constructs two new automata for recognizing, correspondingly, the union and intersection of languages LA and LB$.$

Apply this algorithm to the following two automata:

the automaton from Part $1\mathrm{.}1$ as Automaton A$,$ and

an automaton for words containing only digits as Automaton B$.$

A natural idea is to have $2$ states for Automaton B: words with only digits $($n$),$ and all other words $($w$).$ Start is the starting state, and it is also the only final state. The transitions are as follows:

from n$,$ any digit leads to n$,$ any other symbol leads to w;

from w$,$ every symbol leads back to w$.$

For simplicity, in your automaton for recognizing the union and intersection of the two languages, feel free to assume that you only three symbols $1,$ a$,$ and $?.$