Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-12
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.

The language {01} is not a set of alphabet symbols

Statement refuted

The statement "every language over an alphabet Σ is a subset of Σ" is false.

Facts & Assumptions

Given: The binary alphabet Σ={0,1} and the set L={01}.

[A1]

The statement refuted is: every language over an alphabet Σ is a subset of Σ.

[L1]

A language over Σ is a subset of Σ∗ by Languages over an alphabet.

Counterexample

technique · direct
1.1

The set L={01} is a language over Σ: the unique element 01 is a binary word, so L⊆Σ∗.

givenL1
1.2

But 01∉Σ, because Σ contains only the two one-letter symbols 0 and 1. Therefore L⊈Σ.

given
2.1

So L is a counterexample to [A1].

A1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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