Languages A, B ∈ NP-complete such that A⋃B = Σ* I'm pretty new to complexity theory and it seems like I stuck with this problem. We should find language $B$ such that it accepts any words rejected by $A$ but in that case, it seems that $B$ is a complement of $A$ and therefore $B$ belongs to $coNPC$. What is my mistake? Thank you!
Hint: let $A$ be a NP-complete language and $B=\Sigma^*$.  Then $A \cup B = \Sigma^*$.  Is $B$ the complement of $A$?  Does it belong to coNPC?