Vanishing Ideal of Larger Subset of Affine Space is Smaller

Theorem

Let $k$ be a field.

Let $n \ge 1$ be a natural number.

Let $\mathbb A^n_k$ be the standard affine space over $k$.

Let $S \subseteq T \subseteq \mathbb A^n_k$.

Then:

$\map I S \supseteq \map I T$

where $\map I S$ and $\map I T$ denote the vanishing ideals of $S$ and $T$, respectively.

$\ds f$

$\in$

$\ds \map I T$

$\ds \leadsto \ \ $

$\ds \forall x \in T: \, $

$\ds \map f x$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds \forall x \in S: \, $

$\ds \map f x$

$=$

$\ds 0$

since $S \subseteq T$ by assumption

$\ds \leadsto \ \ $

$\ds f$

$\in$

$\ds \map I S$

$\blacksquare$