CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

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Two disjoint modules whose union is not a module

Statement refuted

If $M$ and $N$ are disjoint modules of a graph, then $M \cup N$ is a module.

Facts & Assumptions

Given: The path $P_{4}$ with vertices $0, 1, 2, 3$ in order, together with the two sets $M = {0}$ and $N = {2}$ .

[L1]

Singletons are trivial modules (Modules of a graph, and the trivial modules).

[L2]

In the path $P_{4}$ , the vertices are adjacent exactly along the edges $01$ , $12$ , and $23$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Counterexample

technique · constructive

1.1givenconstruct

The sets $M = {0}$ and $N = {2}$ are disjoint.

1.2L1

By [L1], both $M$ and $N$ are modules of $P_{4}$ .

2.1step 1.1L2

The vertex $3$ lies outside $M \cup N$ , is adjacent to $2$ , and is not adjacent to $0$ by [L2], so it splits $M \cup N$ . Hence $M \cup N$ is not a module.

3.1step 1.2step 2.1discharge-construct∎

Thus $P_{4}$ with $M$ and $N$ is a counterexample to the claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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.