= Finite-support pointwise operations
{title2=$\operatorname{supp}(F(f,g))\subseteq\operatorname{supp}f\cup\operatorname{supp}g$}
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 $0\leq0$ 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 $f\leq g$.
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