CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-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.
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Without disjoint copies, union does not add generating functions

Counterexample

Let $A = B = {a}$ with $∣ a ∣ = 1$ , and form the ordinary set-theoretic union $A \cup B = {a}$ without adding tags. Then

$OGF (A) = x, OGF (B) = x, OGF (A \cup B) = x \neq 2 x .$

Facts & Assumptions

Given: The symbolic sum rule is proved only for the tagged disjoint union $A + B$ (Disjoint union and Cartesian product translate to addition and multiplication of ordinary generating functions, Disjoint unions and Cartesian products of combinatorial classes).

Verification

technique · direct

1.1

The ordinary union $A \cup B$ has one object of size $1$ , so its generating function is $x$ .

given

2.1

The sum $OGF (A) + OGF (B)$ is $x + x = 2 x$ , which is different from step 1.1. The failure is exactly the missing disjointness: the same object was counted twice on the right and once on the left.

step 1.1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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.