How to understand $I(\emptyset) = k[x_1,\dots,x_n]$?

Question

In our class, for $Z \subseteq \mathbb{A}^n$, we define: $$I(Z) := \{f \in k[x_1,\dots,x_n] \ | f(z) = 0 \ \forall z \in Z\}$$

I understand this as saying that, given a set of points $Z$ in $\mathbb{A}^n$, look for all polynomials that have $Z$ as roots.

For $I(\emptyset)$, the condition $f(z) = 0 \ \forall z \in Z$ is vacuously true for all $f$’s. Is there a way to understand this along the line above?

Answer 1

The condition $f(z)=0$ for all $z\in Z$ means $$\forall z(z\in Z\Rightarrow f(z)=0).$$ In case of $Z=\emptyset$, the premise $z\in Z$ is false and so the implication $(z\in Z\Rightarrow f(z)=0)$ is true for each $z\in Z$, by definition of the truth table of implication. Therefore, $I(\emptyset)$ is as claimed.