 Finite-support pointwise operations

Suppose functions $f,g$ vanish outside finite sets $S_f,S_g$, and a binary operation $F$ satisfies $F(0,0)=0$. The pointwise operation result $h(x)=F(f(x),g(x))$ then vanishes outside the union $S_f\cup S_g$: elsewhere both inputs are zero. If these sets are their supports, this proves the displayed support inclusion.

Enumerating the union is sufficient for addition, subtraction, pointwise maximum and pointwise minimum of signed count functions. It also suffices to test their pointwise order, since $0\le 0$ elsewhere. Taking only the intersection can miss a nonzero value present on just one side. Scanning only the left support can also miss a negative right-side value when checking $f\le g$.