This is from Algebraic Graph theory, by Godsil.
Let $r$ be a function on the subsets of a finite set $\Omega$ and define
$r^*(A)=|A| +r(\Omega \setminus A) - r(\Omega)$
It follows that if $r(\emptyset)=0$ then $(r^*)^*=r$.
I don't see how this follows. $r$ is a function from the subsets of $\Omega$ to nonnegative integers. If I apply $^*$ to a subset of $\Omega$ I get a positive integer and I don't see how I can apply $^*$ to that.
I don't think I understand correctly how the dual function is defined.