I'm currently following a course on Complexity Theory, and whilst studying, I came across a rather counterintuitive statement:
If $\textbf{P}=\textbf{NP}$, the following holds:
For every $A \in \textbf{NP}$, there is a $B \in \textbf{NP}$ such that $A \leq B$ (where $\leq$ means Karp reducible).
However, I do not understand how this applies to the empty problem $\emptyset$, and it's complement $\Sigma^*$, as these only have no-instances and yes-instances, respectively.
Are there other problems in NP such that these two are reducible to them?