Deterministic Finite Automata (DFA) Examples: Sigma*, Empty Set, and More

Here we try to make sense out of various languages, and more importantly, DFAs. The languages we look at, and make DFAs for, are:
1. Sigma*
2. {epsilon}
3. empty set
4. Sigma* \ {epsilon}
5. (empty set)*
6. {epsilon}*

Recall that a DFA is a 5-tuple (Q, Sigma, delta, q0, F) where Q is the set of states, Sigma is the alphabet, delta is the transition function, q0 is the start state, and F is the set of final state. A computation on a string is the sequence of states that are visited, starting from the start state, on that string. A string is accepted if and only if the computation ends in a final state. In the video, a final state is one with a double circle.

▶ADDITIONAL QUESTIONS◀
1. Are these the smallest DFAs for these languages?
2. Is there a set S, not equal to Sigma, where S* = Sigma*?
3. Can we have two DFAs with a different number of states with the same language?

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▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught over 12 courses at Arizona State University, as well as Colgate University, including several sections of undergraduate theory.

[srijitanitamajumdar]

These are the basics that we often find a tad bit confusing while solving problems. Thank you for shedding light on these portions.

[ChrisOffner]

I'm afraid I'm not following the explanation of 5. at around 4:46. Why is the ∅* = {ε}?
You say "If we try to concatenate one or more of anything from the empty set, we're going to get the empty set." at ~4:59.
Not sure how that leads to {ε}.

[arimaJT]

Awesome work, and i love the intro and outro music, always!

[siddharthapurwar]

I really appreciate your helped me a lot