Which strings belong to the regular set represented by the regular expression (1∗01∗0)?

Question

I know the string should be like 1…101…10, but not sure how to describe it. Can anyone help me?

Answer 1

Let $S$ be the regular set represented by the regular expression $1^*01^*0$. Every member of $S$ contains exactly two $0$’s, one of which is the last character of the string. In fact, $S$ is the set of such strings: a string $b_1b_2\ldots b_n$ is in $S$ if and only if $b_n=0$ and exactly one of $b_1,\ldots,b_{n-1}$ is $0$. You could also describe members of $S$ as strings of $0$’s and $1$’s of the form $1^m01^n0$, where $m$ and $n$ may be any non-negative integers.