I've been playing around with the idea of performing logic with implicit equations. For example, given the implicit equations $$\begin{matrix} A:\; \; F(x,y,z,\dots) = F = 0 \\ B:\; \; G(x,y,z,\dots) = G = 0 \end{matrix}$$ we can:
AND: $A \wedge B \equiv {F}^2 + {G}^2 = 0$
OR: $A \vee B \equiv FG = 0$
I was trying to think of a nice way to do NOT and came up with
NOT: $¬A \equiv \lim\frac{\mathcal{O}\left(F_1(x)\right)}{F_1(x)}$
but I feel like there is a smarter way to do it. Any ideas?