Inclusion-exclusion principle

Mathematical statement

For each nonempty index subset I use sign (−1)^(|I|+1): odd-order intersections add and even-order intersections subtract. Equal sets at different indices remain distinct indexed entries, and empty intersections contribute0. For an empty family, union and sum are0. Singles-minus-pairs alone is not a general exact formula for three or more sets; it is a separately identified Bonferroni truncation.

Representative formula (exact TeX)

I(A)

Assumptions

• The complete finite indexed family consists of explicitly listed finite integer sets.
• The general finite-family exercise enumerates every nonempty index subset.

Symbol scope

• I — inclusion-exclusion alternating sum
• A — indexed set family

Intuition, definition, and misconceptions

Every nonempty index subset is required.

Intersection order determines the alternating sign.

Pairwise truncation is not the general exact formula.

For three or more sets, subtracting only pairwise intersections can omit essential higher-order terms.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Set operations — Use the union and every required intersection.
• Basic counting principles — Every term is a finite cardinality.
• Cardinality comparison of sets — The alternating sum is compared with union size.
• Mathematical induction — The general theorem concerns finite indexed families.

Verification boundary

Only finite families are handled; Bonferroni bounds must be identified separately from exact equality.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add generating functions and Catalan counting.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.