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False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31
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False: classical Möbius inversion and inclusion-exclusion are unrelated inversion principles

Statement

Classical divisor Möbius inversion and finite inclusion-exclusion arise from unrelated algebraic principles.

Facts & Assumptions

Given: The divisibility poset and finite Boolean lattices.

[L1]

Classical divisor inversion is lower-finite poset Möbius inversion on positive-integer divisibility (Classical Möbius inversion over positive divisors).

[L2]

Complementary inclusion-exclusion is upper-finite poset Möbius inversion on a finite Boolean lattice (The complementary inclusion-exclusion formula is Möbius inversion on the Boolean lattice).

Refutation

technique · direct
1.1

By [L1], the classical divisor formula is an instance of the general poset inversion theorem.

L1
1.2

By [L2], complementary inclusion-exclusion is another instance of the same theorem, for a different poset.

L2
2.1

Thus the two principles share one incidence-algebra inversion mechanism, so the claim that they are unrelated is false.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 47 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources