ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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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.

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The cosets of $Q$ in $R$ meet $[0, 1]$ in pairwise disjoint classes, and rational translates of a Vitali set count them

Example

Fix a Vitali set $V \subseteq [0, 1]$ . The equivalence classes of $x \sim y ⟺ x - y \in Q$ meet $[0, 1]$ in pairwise disjoint pieces, and the rational translates of $V$ count those classes exactly:

$[0, 1] \subseteq ⋃_{q \in Q \cap [- 1, 1]} (V + q) \subseteq [- 1, 2] .$

Facts & Assumptions

Given: A Vitali set $V \subseteq [0, 1]$ .

[F1]

A Vitali set on $[0, 1]$ meets each class of $x \sim y ⟺ x - y \in Q$ in exactly one point (Vitali set on $[0, 1]$ ).

[L2]

$Q$ is countably infinite ( $Q$ is countably infinite).

Verification

technique · direct

1.1

Two points $x, y \in [0, 1]$ lie in the same class exactly when they differ by a rational, and [F1] says that $V$ contributes one and only one representative to each such class. Thus the pieces $(x + Q) \cap [0, 1]$ are pairwise disjoint and each is hit once by $V$ .

2.1

If $t \in [0, 1]$ , let $v \in V$ be the unique representative of its class. Then $t - v \in Q$ and, because $0 \leq t, v \leq 1$ , also $- 1 \leq t - v \leq 1$ ; so $t \in V + q$ for some $q \in Q \cap [- 1, 1]$ . Conversely every $v + q$ with $v \in V$ and $q \in [- 1, 1]$ lies in $[- 1, 2]$ .

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

Vitali set (Wikipedia) (standard reference, not scraped)

The cosets of QQ in RR meet [0,1][0,1] in pairwise disjoint classes, and rational translates of a Vitali set count them