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 | |||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \map V I\) |
$\blacksquare$