# The situation when P is a superset of NP

Could it be that three languages $$A, B, C$$ such that $$A \subset B \subset C$$, and $$B \in P$$, but $$A$$ and $$C$$ are $$NP$$-complete?

Let $$L$$ be any NP-complete language over $$\Sigma = \{0,1\}$$, and take \begin{align} A &= \{ 0x : x \in L \} \\ B &= 0\Sigma^* \\ C &= 0\Sigma^* \cup \{ 1x : x \in L \} \end{align}
• Should $L$ be unary over $\Sigma = \{0\}$? – Luke Mathieson Dec 2 '20 at 12:54
• How do you get $A \subset B$? – Luke Mathieson Dec 2 '20 at 12:55