All seven intersection terms for three sets

Goal and scope

Audit the full alternating sum and each element's multiplicity.

A finite indexed family, retaining separate indices for repeated sets.

Givens and symbols

• work

• G1 — A={1,2,3}, B={2,3,4}, C={3,4,5}.

• r — number of indexed sets containing an element

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — 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 contribute 0. For an empty family, the union is empty and both its cardinality and the inclusion-exclusion sum are 0. Singles-minus-pairs alone is not a general exact formula for three or more sets; it is a separately identified Bonferroni truncation. · mko-inclusion-exclusion-principle
• work · prerequisite — Set unions and intersections, finite summation by element multiplicity; the binomial identity(1-1)^r=sum_j(-1)^j comb(r,j) is an explicit prerequisite.

Main derivation

A\cup B\cup C

\{1,2,3,4,5\},\quad|A\cup B\cup C|=5

|A|,|B|,|C|

3,3,3

A\cap B,A\cap C,B\cap C

\{2,3\},\{3\},\{3,4\}

A\cap B\cap C

\{3\},\quad|A\cap B\cap C|=1

3+3+3-2-1-2+1

5=|A\cup B\cup C|

x\text{ belongs to }r\ge1\text{ indexed sets}

\sum_{j=1}^{r}(-1)^{j+1}\binom rj=1-(1-1)^r=1

\text{singles minus pairs only}

9-5=4\ne5

(A,A)\text{ indexed twice};\quad n=0

|A|+|A|-|A|=|A|;\quad|\bigcup\varnothing|=0

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — Equal sets do not mean an indexed obligation may be ignored.
• S07 · premise_audit · G1, S06, work — A Bonferroni bound must be separately labeled, not asserted as equality.
• S06 · premise_audit · G1, S05, work — Elements outside the union have r=0 and contribute zero separately.
• S05 · premise_audit · G1, S04, work — Three singles, three pairs and one triple equal2³-1.
• S04 · premise_audit · G1, S03, work — Listing pairwise intersections does not permit omitting the triple.
• S03 · premise_audit · G1, S02, work — Second-order terms carry negative signs.
• S02 · premise_audit · G1, S01, work — Each indexed entry is counted once.
• S01 · premise_audit · G1, work — The expected result is not merely assumed from the formula.

Error controls and negative controls

Seven terms

Enumerate index subsets, not distinct set values.

Elementwise

Use1-(1-1)^r for r≥1 and zero separately for r=0.

• Singles minus pairs is exact for three sets. → reject — Here4 differs from union size 5.
• The triple intersection should be subtracted. → reject — Order three is odd and has positive sign.
• Repeated sets make the union twice as large. → reject — The second-order intersection subtracts the duplicate copy.

Result and limitations

All seven terms give union size 5; elementwise multiplicity proves the finite-family formula.

Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: ea75129552baa4d33afae410265a8e2ec2eda5b1a8669c83da8afed4efc43c95
• original source SHA-256: d12d13f29453c49f15f41deea45201369e57e7476d15786a97750ca414dc608a
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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