What does {a,b}* for DFA's mean?

For instance when the question contains $$\{a,b\}^*$$ does this mean that the DFA must have at least one $$a$$ and one $$b$$ on top of whatever conditions it has?
For example a DFA that accepts $$\{w \in \{a,b\}^* : w \text{ contains } bbb\}$$ should it reject the actual string $$bbb$$ because it does not contain an $$a$$?

• What definition of $*$ do you use? – greybeard Feb 5 '20 at 8:39

$$\{a,b\}$$ is set notation. The alphabet of a language is the set of all possible symbol used in the language. You can substitute the typical $$\Sigma$$ for the full set every time.
$$\Sigma^*$$ (where $$^*$$ is the kleene star) is the set of all possible strings you can make with that alphabet. That includes strings that happen to not use a certain character.
Therefor $$\{w \in \{a,b\}^* : w \text{ contains } bbb\}$$ is equivalent to $$\{w \in \Sigma^* : \Sigma = \{a,b\} \text{ and } w \text{ contains } bbb\}$$. But it skipped defining $$\Sigma$$ for terseness reasons.