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