$1\mathrm{.}1.$ Use the same ideas to describe an automaton for homework numbers. A homework number should start with a digits, it can be followed by digits and letters. For example, $12$ and $12$a should be accepted, while a$1$ should be rejected.

A natural idea is to have $3$ states:

the start state S$,$

the state V indicating we have a valid homework number, and

the state E indicating that this is an error.

Start is the starting state, V is the only final state. The transitions are as follows:

from S$,$ any digits lead to state V$,$ any other letter leads to state E;

from V$,$ any digits and any letter letter lead to state V$,$ any other symbol $($e$.$g$.,?)$ leads to state E;

from E$,$ any symbol leads back to E$.$

$1\mathrm{.}2.$ Trace, step$-$by$-$step, how the finite automaton from Part $1\mathrm{.}1$ will check whether the following two words $($sequences of symbols$)$ are valid homework numbers:

the word $1$a $($which this automaton should accept$)$ and

the word a$1($which this automaton should reject$).$

$1\mathrm{.}3.$ Write down the tuple corresponding to the automaton from Part $1\mathrm{.}1$:

Q is the set of all the states,

$\backslash$Sigma is the alphabet, i$.$e$.,$ the set of all the symbols that this automaton can encounter; for simplicity, consider only three symbols: $1,$ a$,$ and question mark $?$;

$\backslash$delta : Q x $\backslash$Sigma $->$ Q is the function that describes, for each state q and for each symbol s$,$ the state $\backslash$delta $($q$,$s$)$ to which the automaton that was originally in the state q moves when it sees the symbol s $($you do not need to describe all possible transitions this way, just describe two of them$)$;

q$0$ is the starting state, and

F is the set of all final states.

$1\mathrm{.}4.$ 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 $?.$