Construction of DFA using an odd bit of language

Question

I am working through a lecture and it constructs a DFA using the language:

$$\{w\mid w\textsf{ is any string not in }(ab^+)^\ast\}$$

What does the $(ab^+)$ mean?

Answer 1

The expression $b^+$ is an abbreviation for $bb^*$: it stands for any string of one or more $b$s. Thus, $ab^+$ matches any string of the form $ab^n$ with $n\ge 1$, and $(ab^+)^*$ matches any concatenation of $0$ or more such strings. Typical words of this language are $\lambda$ (the empty word – you may use the symbol $\epsilon$), $abb$, $ababbbababbbbbb$, $abababb$, and so on. In other words, every $a$ is followed by at least one $b$, and the first character (if the word is non-empty) must be $a$.