# Question 20937: A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula [Return to catalogue page 105](https://ocme.evemisslab.com/en/questions/page/105/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations/#q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0087) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0040-20937 · q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0087 ## Original question and givens Use the complete indexed finite integer-set family sets=[[11, 12, 13], [11, 13, 14], [11, 13, 15]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary). - G001 — Use the complete indexed finite integer-set family sets=[[11, 12, 13], [11, 13, 14], [11, 13, 15]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-inclusion-exclusion-principle](https://ocme.evemisslab.com/en/mko/mko-inclusion-exclusion-principle/) ## Step-by-step derivation S001 · impliesBefore source operation=incexc-pairwise-only-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary] Conditions: none Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL indexed finite set family in SUPPLIED outer index order=[[11, 12, 13], [11, 13, 14], [11, 13, 15]] After within-set numeric normalization=[[11, 12, 13], [11, 13, 14], [11, 13, 15]];duplicates INSIDE a set rejected;EQUAL sets at DIFFERENT indices RETAINED;integer labels may be negative Conditions: none Normalize elements within each set only.Outer family positions are inclusion-exclusion indices and must not be semantically reordered or deduplicated.Empty sets are legitimate indexed members,and element labels are not cardinalities. Step identifiers and dependenciesRule: incexc_indexed_family_not_deduplicated_or_semantic_reordered · Uses: G001, S001 S003 · calculationBefore inclusion mask1 signed intersection cardinality := mul(1, 3) After inclusion mask1 signed intersection cardinality = 3 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore inclusion mask1 alternating prefix sum := sum(0, 3) After inclusion mask1 alternating prefix sum = 3 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore next NONEMPTY NUMERIC bitmask=1;ORIGINAL family indices=[0];actual selected original sets=[[11, 12, 13]] After FULL intersection members=[11, 12, 13];size=3;ODD order sign=1;alternating prefix=3;zero-size terms retained Conditions: none Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality. Step identifiers and dependenciesRule: incexc_every_index_submask_intersection_parity_source_order · Uses: G001, S004 S006 · calculationBefore inclusion mask2 signed intersection cardinality := mul(1, 3) After inclusion mask2 signed intersection cardinality = 3 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore inclusion mask2 alternating prefix sum := sum(3, 3) After inclusion mask2 alternating prefix sum = 6 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore next NONEMPTY NUMERIC bitmask=2;ORIGINAL family indices=[1];actual selected original sets=[[11, 13, 14]] After FULL intersection members=[11, 13, 14];size=3;ODD order sign=1;alternating prefix=6;zero-size terms retained Conditions: none Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality. Step identifiers and dependenciesRule: incexc_every_index_submask_intersection_parity_source_order · Uses: G001, S007 S009 · calculationBefore inclusion mask3 signed intersection cardinality := mul(-1, 2) After inclusion mask3 signed intersection cardinality = -2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore inclusion mask3 alternating prefix sum := sum(6, -2) After inclusion mask3 alternating prefix sum = 4 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore next NONEMPTY NUMERIC bitmask=3;ORIGINAL family indices=[0, 1];actual selected original sets=[[11, 12, 13], [11, 13, 14]] After FULL intersection members=[11, 13];size=2;EVEN order sign=-1;alternating prefix=4;zero-size terms retained Conditions: none Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality. Step identifiers and dependenciesRule: incexc_every_index_submask_intersection_parity_source_order · Uses: G001, S010 S012 · calculationBefore inclusion mask4 signed intersection cardinality := mul(1, 3) After inclusion mask4 signed intersection cardinality = 3 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore inclusion mask4 alternating prefix sum := sum(4, 3) After inclusion mask4 alternating prefix sum = 7 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore next NONEMPTY NUMERIC bitmask=4;ORIGINAL family indices=[2];actual selected original sets=[[11, 13, 15]] After FULL intersection members=[11, 13, 15];size=3;ODD order sign=1;alternating prefix=7;zero-size terms retained Conditions: none Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality. Step identifiers and dependenciesRule: incexc_every_index_submask_intersection_parity_source_order · Uses: G001, S013 S015 · calculationBefore inclusion mask5 signed intersection cardinality := mul(-1, 2) After inclusion mask5 signed intersection cardinality = -2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore inclusion mask5 alternating prefix sum := sum(7, -2) After inclusion mask5 alternating prefix sum = 5 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore next NONEMPTY NUMERIC bitmask=5;ORIGINAL family indices=[0, 2];actual selected original sets=[[11, 12, 13], [11, 13, 15]] After FULL intersection members=[11, 13];size=2;EVEN order sign=-1;alternating prefix=5;zero-size terms retained Conditions: none Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality. Step identifiers and dependenciesRule: incexc_every_index_submask_intersection_parity_source_order · Uses: G001, S016 S018 · calculationBefore inclusion mask6 signed intersection cardinality := mul(-1, 2) After inclusion mask6 signed intersection cardinality = -2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore inclusion mask6 alternating prefix sum := sum(5, -2) After inclusion mask6 alternating prefix sum = 3 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore next NONEMPTY NUMERIC bitmask=6;ORIGINAL family indices=[1, 2];actual selected original sets=[[11, 13, 14], [11, 13, 15]] After FULL intersection members=[11, 13];size=2;EVEN order sign=-1;alternating prefix=3;zero-size terms retained Conditions: none Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality. Step identifiers and dependenciesRule: incexc_every_index_submask_intersection_parity_source_order · Uses: G001, S019 S021 · calculationBefore inclusion mask7 signed intersection cardinality := mul(1, 2) After inclusion mask7 signed intersection cardinality = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore inclusion mask7 alternating prefix sum := sum(3, 2) After inclusion mask7 alternating prefix sum = 5 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore next NONEMPTY NUMERIC bitmask=7;ORIGINAL family indices=[0, 1, 2];actual selected original sets=[[11, 12, 13], [11, 13, 14], [11, 13, 15]] After FULL intersection members=[11, 13];size=2;ODD order sign=1;alternating prefix=5;zero-size terms retained Conditions: none Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality. Step identifiers and dependenciesRule: incexc_every_index_submask_intersection_parity_source_order · Uses: G001, S022 S024 · calculationBefore inclusion COMPLETE actual union cardinality := sum(1, 1, 1, 1, 1) After inclusion COMPLETE actual union cardinality = 5 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore ACTUAL union element=11;ALL containing ORIGINAL family indices=[0, 1, 2];indexed multiplicity r=3 After only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1 Conditions: none Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated. Step identifiers and dependenciesRule: incexc_each_element_full_coefficient_not_one_sum_coincidence · Uses: G001, S024 S026 · impliesBefore ACTUAL union element=12;ALL containing ORIGINAL family indices=[0];indexed multiplicity r=1 After only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1 Conditions: none Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated. Step identifiers and dependenciesRule: incexc_each_element_full_coefficient_not_one_sum_coincidence · Uses: G001, S025 S027 · impliesBefore ACTUAL union element=13;ALL containing ORIGINAL family indices=[0, 1, 2];indexed multiplicity r=3 After only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1 Conditions: none Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated. Step identifiers and dependenciesRule: incexc_each_element_full_coefficient_not_one_sum_coincidence · Uses: G001, S026 S028 · impliesBefore ACTUAL union element=14;ALL containing ORIGINAL family indices=[1];indexed multiplicity r=1 After only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1 Conditions: none Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated. Step identifiers and dependenciesRule: incexc_each_element_full_coefficient_not_one_sum_coincidence · Uses: G001, S027 S029 · impliesBefore ACTUAL union element=15;ALL containing ORIGINAL family indices=[2];indexed multiplicity r=1 After only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1 Conditions: none Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated. Step identifiers and dependenciesRule: incexc_each_element_full_coefficient_not_one_sum_coincidence · Uses: G001, S028 S030 · impliesBefore COMPLETE indexed family size=3;ALL7 nonempty index subsets accounted for;ACTUAL union=[11, 12, 13, 14, 15] After each covered element has coefficient1 and any uncovered element coefficient0;FULL alternating sum=5=actual union size=5;empty family yields empty union/no terms/0 Conditions: none Sum the pointwise coefficient certificates to get the entire finite union size,rather than inferring a theorem from one numerical coincidence.Uncovered points have no containing indices and contribute0;the empty family gives0 directly. Step identifiers and dependenciesRule: incexc_full_pointwise_identity_all_finite_indexed_sets · Uses: G001, S029 S031 · calculationBefore inclusion pairwise truncation singleton sum := sum(3, 3, 3) After inclusion pairwise truncation singleton sum = 9 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore inclusion pairwise truncation ALL pair intersections := sum(2, 2, 2) After inclusion pairwise truncation ALL pair intersections = 6 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore inclusion singles MINUS pairs truncation := sub(9, 6) After inclusion singles MINUS pairs truncation = 3 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore inclusion true union MINUS pairwise truncation := sub(5, 3) After inclusion true union MINUS pairwise truncation = 2 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · impliesBefore ALL singleton sizes sum=9;ALL pair intersection sizes=[2, 2, 2];pair sum=6;actual union size=5 After pairwise truncation=3;numerically equals THIS union=No;GENERAL pairwise-only exactness claim=No;Bonferroni lower-bound gap=2>=0,not a cardinality assertion for the truncation Conditions: none At r=0 both contributions are0;for r≥1 the truncated coefficient r−C(r,2)≤1,with equality for r=1,2 and a strict gap for r≥3.So the total is a lower bound and may be negative.This instance may coincide numerically;the final No still withholds a general exact formula without negating the equality flag. Step identifiers and dependenciesRule: incexc_pairwise_No_not_numerical_inequality_or_nonnegative_count · Uses: G001, S034 ## Reverse checks - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Recompute inclusion true union MINUS pairwise truncation with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute inclusion singles MINUS pairs truncation with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute inclusion pairwise truncation ALL pair intersections with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute inclusion pairwise truncation singleton sum with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Recompute inclusion COMPLETE actual union cardinality with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Recompute inclusion mask7 alternating prefix sum with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute inclusion mask7 signed intersection cardinality with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute inclusion mask6 alternating prefix sum with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute inclusion mask6 signed intersection cardinality with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute inclusion mask5 alternating prefix sum with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute inclusion mask5 signed intersection cardinality with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute inclusion mask4 alternating prefix sum with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute inclusion mask4 signed intersection cardinality with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute inclusion mask3 alternating prefix sum with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute inclusion mask3 signed intersection cardinality with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute inclusion mask2 alternating prefix sum with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute inclusion mask2 signed intersection cardinality with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute inclusion mask1 alternating prefix sum with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute inclusion mask1 signed intersection cardinality with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: a8b6c83281e592a6a7a255d365fedd16e465f33cf685f7a6bb1ac1e218827542 - candidate record SHA-256: b4c0af3b6c6ea48a0426b72a2088ce658b5da5f374d974a34c30b2ff493d7b70 - semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d - question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0087.json)