Zero Locus of Larger Set is Smaller

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Theorem

Let $k$ be a field.

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

Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomials in $n$ variables over $k$.

Let $I, J \subseteq A$ be subsets, and $\map V I$ and $\map V J$ their zero loci.

Let $I \subseteq J$.


Then $\map V I \supseteq \map V J$.


Proof

\(\ds x\) \(\in\) \(\ds \map V J\)
\(\ds \leadsto \ \ \) \(\ds \forall f \in J: \, \) \(\ds \map f x\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \forall f \in I: \, \) \(\ds \map f x\) \(=\) \(\ds 0\) since $I \subseteq J$ by assumption
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \map V I\)

$\blacksquare$