DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-26

How 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.

Vitali set on $[0, 1]$

Definition

Define an equivalence relation on $[0, 1]$ by

$x \sim y : ⟺ x - y \in Q .$

A subset $V \subseteq [0, 1]$ is a Vitali set on $[0, 1]$ when every equivalence class of this relation meets $V$ in exactly one point. Equivalently, for every $x \in [0, 1]$ there is a unique $v \in V$ with $x - v \in Q$ .

Remarks

The definition names a Vitali set, not the Vitali set. Different choice functions produce different selectors.
The relation is taken on $[0, 1]$ , not on all of $R$ , because the later countable-translate argument only needs one selector on one bounded interval.

Depends on

Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length

Used by

Dependency tree · two levels

5 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

Jacek Cichoń, Aleksander Kharazishvili, and Bogdan Węglorz, Subsets of the Real Line, Chapter 8 (standard reference, not scraped)
Vitali set (Wikipedia) (standard reference, not scraped)

Vitali set on [0,1][0,1]