PS2, Part (h)

[lucashkaplan]

Hey all, I had a question regarding PS2, part (h). Everything in the solution makes sense to me, except for one part. I don't understand why if a language S is boring, then so is NOT(S). Any explanation would be greatly appreciated.

[spamegg1]

Definition: $S$ is boring when either $S$ or $\overline{S}$ is 0-finite.

1. Assume $S$ is boring.
   [We want to show $\overline{S}$ is boring. In other words we want to show either $\overline{S}$ or $\overline{\overline{S}}$ is 0-finite.]
2. By 1 and the definition, either $S$ or $\overline{S}$ is 0-finite.
3. By 2, either $\overline{\overline{S}}$ or $\overline{S}$ is 0-finite. [Because $S = \overline{\overline{S}}$.]
4. Rewriting 3, either $\overline{S}$ or $\overline{\overline{S}}$ is 0-finite.
5. By 4 and the definition, $\overline{S}$ is boring.