Differentiation of Polynomials induces Well-Founded Relation

Theorem

Let $P$ be the set of all polynomials over $\R$ in one variable with real coefficients.

Let $\DD$ be a relation on $P$ defined as:

$\forall p_0, p_1 \in P: \tuple {p_0, p_1} \in \DD$ if and only if $p_0$ is the derivative of $p_1$.

Then $\DD$ is a well-founded relation on $P$.

Proof

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Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $7$