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Let $\Sigma = \{a, b\}$ and $L = \{aa, bb\}$. Use set notation to describe $\overline L$.

This is exercise 6 (page 28) from "An Introduction to Formal Languages and Automata" by Peter Linz. The author provides the following answer: $\overline L = \{\lambda, a, b, ab, ba\} \cup \{ w \in \{a, b\}^+ : |w|\geq 3\}$. Mine on the other hand is the following: $\overline L = \Sigma^* - L = \{w \in \Sigma^* : w \neq aa, w \neq bb\}$. Is my answer wrong? Thanks in advance.

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    $\begingroup$ What is meant by "set notation"? Your answer is what I would write. Anyway, representation is definitely not unique and I wouldn't care that much about it. $\endgroup$ – user114966 Feb 14 at 21:40
  • $\begingroup$ Your answer is not wrong, but the book's answer makes it easier to write a regular expression or regular grammar. $\endgroup$ – rici Feb 14 at 21:55
  • $\begingroup$ Alright then. Thanks! $\endgroup$ – Mauricio Mendes Feb 14 at 22:01
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You already got your answer in the comments: both the solutions provided solve the task. The only thing in which they differ is from which angle the attempt to solve the task. The author one goes from an empty set and adds only words that satisfy the given constraints (bottom-up approach). Yours goes from the set of all possible words and removes the ones that do not satisfy the constraint (top-down approach).

I would say his is more accurate IF you are not allowed to use $L$ or $\Sigma$, as it is often the case in exercise text books.

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