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The complementary inclusion-exclusion formula is Möbius inversion on the Boolean lattice

Statement

For a finite sieve family $(A_i)_{i\in I}$ in a finite ambient set $X$ , let $\iota:\mathbb N\to\mathbb R$ be the canonical natural map used in Inclusion and exclusion: $\iota\lvert\bigcup_{i \in I} A_i\rvert = \sum_{\varnothing \ne J \subseteq I}(-1)^{\lvert J\rvert + 1}\,\iota\lvert A_J\rvert$ , together with the complementary form counting the elements in none of the $A_i$ . The complementary inclusion-exclusion identity

$\iota|X\setminus\bigcup_{i\in I}A_i|=\sum_{J\subseteq I}(-1)^{|J|}\,\iota|A_J|$

in $\mathbb R$ is exactly the upper-finite Möbius inversion formula on the Boolean lattice $\mathcal P(I)$ .

Facts & Assumptions

Given: A sieve family $X,I,(A_i)_{i\in I}$ , its intersections $A_J$ , and the trace $T(x):=\{i\in I:x\in A_i\}$ (A finite family $(A_i)_{i \in I}$ of subsets of a finite set $X$ , the intersections $A_J$ for $J \subseteq I$ , and the convention $A_\varnothing = X$ ).

[F1]

For $J\subseteq I$ , $x\in A_J$ exactly when $J\subseteq T(x)$ , including $A_\varnothing=X$ ; and $x\notin\bigcup_iA_i$ exactly when $T(x)=\varnothing$ (A finite family $(A_i)_{i \in I}$ of subsets of a finite set $X$ , the intersections $A_J$ for $J \subseteq I$ , and the convention $A_\varnothing = X$ ).

[L1]

Both forms of Möbius inversion hold on the finite Boolean lattice (Both forms of Möbius inversion hold on every finite poset).

[L2]

Its Möbius function is $\mu(J,K)=(-1)^{|K\setminus J|}$ (For $A\subseteq B$ in a finite Boolean lattice, $\mu(A,B)=(-1)^{\lvert B\setminus A\rvert}$ ).

[L3]

The complementary inclusion-exclusion theorem uses the canonical map $\iota:\mathbb N\to\mathbb R$ , has the displayed terms, and adopts the convention $A_\varnothing=X$ (Inclusion and exclusion: $\iota\lvert\bigcup_{i \in I} A_i\rvert = \sum_{\varnothing \ne J \subseteq I}(-1)^{\lvert J\rvert + 1}\,\iota\lvert A_J\rvert$ , together with the complementary form counting the elements in none of the $A_i$ ).

Proof

technique · direct

1.1

For $K\subseteq I$ , let $h(K):=\iota|\{x\in X:T(x)=K\}|$ , and let $g(J):=\iota|A_J|$ . The trace classes partition $A_J$ , and [F1] gives $g(J)=\sum_{K\supseteq J}h(K)$ in $\mathbb R$ .

F1L3

2.1

Apply the upper-finite form of [L1] at $J=\varnothing$ : $h(\varnothing)=\sum_{K\supseteq\varnothing}\mu(\varnothing,K)g(K)$ .

step 1.1L1

3.1

By [F1], $h(\varnothing)=\iota|X\setminus\bigcup_iA_i|$ , and by [L2], $\mu(\varnothing,K)=(-1)^{|K|}$ . Substitution in step 2.1 gives $\iota|X\setminus\bigcup_iA_i|=\sum_{K\subseteq I}(-1)^{|K|}\,\iota|A_K|$ .

step 2.1F1L2

4.1

The identity in step 3.1 matches [L3] term for term, including the empty-subset term $\iota|A_\varnothing|=\iota|X|$ ; hence complementary inclusion-exclusion is Boolean-lattice Möbius inversion.

step 3.1L3∎

Depends on

Used by

False: classical Möbius inversion and inclusion-exclusion are unrelated inversion principles False statement

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Sources

R. Stanley, Enumerative Combinatorics, Volume 1, §§3.6–3.8 (standard reference, not scraped)