Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
A set family whose incidence vectors are dependent over and independent over
Statement refuted
Changing the field can change linear independence. Take
Facts & Assumptions
Given: the three subsets above.
Their incidence vectors are , and (The incidence vector of a subset over a stated field).
Counterexample
Over , the three incidence vectors sum to , so they are linearly dependent.
Over , suppose . The three coordinates give , , and . Hence and the last equation gives , so . Thus the vectors are linearly independent over .
Depends on
- The incidence vector $v_A\in F^{n}$ of a subset $A\subseteq[n]$ over a stated field
- A finite family of subsets of $[n]$ and its incidence matrix over $F$
- Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i<n} \lambda_i v_i = 0_V$ forces every $\lambda_i = 0_F$, and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent
- For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Matousek, Thirty-three Miniatures, Miniature 3 (standard reference, not scraped)