Does $\{0, 1\}^*\in \text{co-NP}$? There is trivially $\emptyset\in\text{NP}$. From the definition of $\text{co-NP} = \{L : \overline{L} \in \text{NP}\}$ where $\overline{L} = \{0, 1\}^*-L$ follows $\{0, 1\}^*\in \text{co-NP}$. Is this true? If so, it is also true that $\{0,1\}^*\in \text{NP}$?
Both $\emptyset$ and $\{0,1\}^*$ are in $P$, and therefore they are also in $NP \supseteq P$ and $co-NP \supseteq P$.