I have a homework question that asks to draw the transition diagram for the following:
Draw transition diagrams for the DFAs below for the following languages. The alphabet Σ in this example is {0,1}.
1) {p001q | p,q ∈ Σ*}.
2) {λ} ---> λ is an empty string.
The questions I have are, for question 1, will p and q take different values? What I mean is, since different variables are used, does that mean the DFA will only accept strings where and first and last value are different, and the 3 middle values must be 001?
And for question 2, I think this will just be one accepting state that has one loop back to itself.
The first question has been dealt with in the comments.
The DFA with a single state and transitions that loop at that state treats every $w\in\Sigma^*$ identically: if the state is an acceptor state, it accepts every word in $\Sigma^*$, and if the state is not an acceptor state, it accepts no word at all. Thus, you get either $\Sigma^*$ or $\varnothing$, not $\{\lambda\}$.
To get $\{\lambda\}$, you need to be able to distinguish between empty input and non-empty input; this requires a minimum of two states. If $q_0$ is your initial state, you need another state $q_1$. Clearly $q_0$ should be an acceptor state, so that you accept the empty word, and any other input at all must take you out of $q_0$. Thus, you want transitions
$$q_0\overset{0,1}\longrightarrow q_1\;.$$
Moreover, once you get to $q_1$, you never want to return to $q_0$, so you also want transitions
$$q_1\overset{0,1}\longrightarrow q_1\;.$$
In short, $q_1$ is a non-accepting trap state – what I often call the garbage state.