Regular and Non-Regular Language

Question

My friends and I are taking a formal languages class and for one of our homework questions we have to prove if these languages are regular:

1) L = {a^pa^qi : p and q are fixed integer values, i >= 0}

2) L = {a^pb^q : p and q are fixed integer values}

For (1), we know it is regular because we can make a DFA out of it, but what we are not sure how exactly the DFA should look like. Should it not be consist only of one state, say q₀, and it acts as both the initial and final state, with a loop of "a" moves going through it? Our line of thought is because p and q could be any integers, so whatever p and q would be, we would always end up with a certain number of a's, but we are not sure we are 100% correct.

For number (2), we say it is not regular because p and q could be infinite, thus making the language automatically irregular.

We are still quite confused on which concepts are correct. This is still a very abstract concept for all of us and any guidance would be appreciated. Thank you!

Answer 1

Think of it this way. Fix $p$ and $q$ arbitrarily – for example, take $p = 4$ and $q = 5$. Then $$ \begin{align*} L_1 &= \{ a^4 b^{5i} : i \geq 0 \}, \\ L_2 &= \{ a^4 b^5 \}. \end{align*} $$ Now try to answer your question. The answer will be the same no matter what values you choose for $p$ and $q$.