False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

FALSE: a language is a set of symbols

Statement

False claim: if $Σ$ is an alphabet, then every language over $Σ$ is a subset of $Σ$ .

Facts & Assumptions

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

[A1]

The false claim: 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.

[L2]

The word $01$ is a binary word of length $2$ , hence an element of $Σ^{*}$ , by Computation alphabets, words, the empty word, and $Σ^{*}$ .

Refutation

technique · direct

1.1

By [L2], the word $01$ lies in $Σ^{*}$ , so the one-element set $L = {01}$ is a subset of $Σ^{*}$ . Therefore [L1] shows that $L$ is a language over $Σ$ .

givenL1L2

1.2

The element $01$ is a word of length $2$ , not a single symbol of the alphabet $Σ = {0, 1}$ . So $01 \notin Σ$ , and therefore $L$ is not a subset of $Σ$ .

givenL2

2.1

Thus $L$ is a language over $Σ$ for which the conclusion of [A1] fails. The claim is false.

A1step 1.1step 1.2∎

Depends on

Used by

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

Dependency tree · two levels

6 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

Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)
Michael Sipser, MIT 18.404J Theory of Computation, Lecture 1 (standard reference, not scraped)