Regular expressions for some languages

Question

1. Set of strings over $\{0,1\}$ having at least two occurrence of the substring 00.
2. $\{a^n b^m : n ≥ 4, m ≥ 3\}$.
3. Set of strings over the alphabet $\{a,b,c\}$ containing at least one $a$ and one $b$.

Answer 1

Well, think about what's necessary in each case.

In the first case, you need two instances of $00$, and you don't care what else is there. If those instances can't overlap, this would be $(\cdot^*) 00 (\cdot^*) 00 (\cdot^*)$, since $(\cdot^*)$ means "anything at all, including nothing". If the instances can overlap, you should union in $(\cdot^*)000(\cdot^*)$, to catch cases like $000$.

In the second case, you need at least four $n$s, then at least three $m$s. So just specify that: $nnnn(n^*) mmm(m^*)$.