 Solution

Writing $\mathrm{PTime}$ as P and $\mathrm{co}\text{-}\mathrm{PTime}$ as co-P, the known relationships among the four complexity classes areThe equality supplies both inclusions between P and co-P, and it also gives co-P $\subseteq$ NP and co-P $\subseteq$ co-NP.

For the equality, a deterministic polynomial-time algorithm halts on every input; interchange its accepting and rejecting outcomes to decide the complement of a set in the same polynomial time. For P $\subseteq$ NP, a Nondeterministic Turing machine can simply make no choices and follow the deterministic computation. Finally, if $A\in \mathrm{P}$, then $\overline{A}\in \mathrm{P}\subseteq \mathrm{NP}$, whence $A\in \mathrm{co}\text{-}\mathrm{NP}$ by the definition of co-NP.

Neither NP $\subseteq$ co-NP nor co-NP $\subseteq$ NP is known; either would imply equality by taking complements. Nor is either inclusion of P into NP or co-NP known to be strict. These unresolved relationships must not be presented as strict separations.