False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

FALSE: the union of two sigma-algebras on one set is a sigma-algebra

Statement

The union of any two sigma-algebras on the same set is a sigma-algebra.

Facts & Assumptions

Given: The set $X : = {1, 2, 3, 4}$ , $A : = {1, 2}$ , and $B : = {1, 3}$ .

[L1]

A sigma-algebra is closed under finite intersections (Sigma-algebras).

Refutation

technique · counterexample

1.1L1algebra

The families $A : = {\emptyset, X, A, X ∖ A}$ and $B : = {\emptyset, X, B, X ∖ B}$ each satisfy the sigma-algebra axioms.

2.1step 1.1L1∎

Their union contains $A$ and $B$ but not $A \cap B = {1}$ . It therefore fails the closure in [L1] and is not a sigma-algebra.

Depends on

Sigma-algebras

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Exercise 2.2 (standard reference, not scraped)