Question 20924: All Nonempty Index Subsets Are Needed for the Exact Alternating Sum

Original question and givens

Use the complete indexed finite integer-set family sets=[[23, 24, 25], [23, 25, 26], [23, 25, 27], [23, 28]] 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 (family_size, terms, union_size, alternating_sum, equal, certificate).

• G001 — Use the complete indexed finite integer-set family sets=[[23, 24, 25], [23, 25, 26], [23, 25, 27], [23, 28]] 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 (family_size, terms, union_size, alternating_sum, equal, certificate).

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

Step-by-step derivation

source operation=incexc-general-alternating-sum;original snapshot and mathematics retained

explicit ordered source output fields=[family_size, terms, union_size, alternating_sum, equal, certificate]

ORIGINAL indexed finite set family in SUPPLIED outer index order=[[23, 24, 25], [23, 25, 26], [23, 25, 27], [23, 28]]

within-set numeric normalization=[[23, 24, 25], [23, 25, 26], [23, 25, 27], [23, 28]];duplicates INSIDE a set rejected;EQUAL sets at DIFFERENT indices RETAINED;integer labels may be negative

inclusion mask1 signed intersection cardinality := mul(1, 3)

inclusion mask1 signed intersection cardinality = 3

inclusion mask1 alternating prefix sum := sum(0, 3)

inclusion mask1 alternating prefix sum = 3

next NONEMPTY NUMERIC bitmask=1;ORIGINAL family indices=[0];actual selected original sets=[[23, 24, 25]]

FULL intersection members=[23, 24, 25];size=3;ODD order sign=1;alternating prefix=3;zero-size terms retained

inclusion mask2 signed intersection cardinality := mul(1, 3)

inclusion mask2 signed intersection cardinality = 3

inclusion mask2 alternating prefix sum := sum(3, 3)

inclusion mask2 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=2;ORIGINAL family indices=[1];actual selected original sets=[[23, 25, 26]]

FULL intersection members=[23, 25, 26];size=3;ODD order sign=1;alternating prefix=6;zero-size terms retained

inclusion mask3 signed intersection cardinality := mul(-1, 2)

inclusion mask3 signed intersection cardinality = -2

inclusion mask3 alternating prefix sum := sum(6, -2)

inclusion mask3 alternating prefix sum = 4

next NONEMPTY NUMERIC bitmask=3;ORIGINAL family indices=[0, 1];actual selected original sets=[[23, 24, 25], [23, 25, 26]]

FULL intersection members=[23, 25];size=2;EVEN order sign=-1;alternating prefix=4;zero-size terms retained

inclusion mask4 signed intersection cardinality := mul(1, 3)

inclusion mask4 signed intersection cardinality = 3

inclusion mask4 alternating prefix sum := sum(4, 3)

inclusion mask4 alternating prefix sum = 7

next NONEMPTY NUMERIC bitmask=4;ORIGINAL family indices=[2];actual selected original sets=[[23, 25, 27]]

FULL intersection members=[23, 25, 27];size=3;ODD order sign=1;alternating prefix=7;zero-size terms retained

inclusion mask5 signed intersection cardinality := mul(-1, 2)

inclusion mask5 signed intersection cardinality = -2

inclusion mask5 alternating prefix sum := sum(7, -2)

inclusion mask5 alternating prefix sum = 5

next NONEMPTY NUMERIC bitmask=5;ORIGINAL family indices=[0, 2];actual selected original sets=[[23, 24, 25], [23, 25, 27]]

FULL intersection members=[23, 25];size=2;EVEN order sign=-1;alternating prefix=5;zero-size terms retained

inclusion mask6 signed intersection cardinality := mul(-1, 2)

inclusion mask6 signed intersection cardinality = -2

inclusion mask6 alternating prefix sum := sum(5, -2)

inclusion mask6 alternating prefix sum = 3

next NONEMPTY NUMERIC bitmask=6;ORIGINAL family indices=[1, 2];actual selected original sets=[[23, 25, 26], [23, 25, 27]]

FULL intersection members=[23, 25];size=2;EVEN order sign=-1;alternating prefix=3;zero-size terms retained

inclusion mask7 signed intersection cardinality := mul(1, 2)

inclusion mask7 signed intersection cardinality = 2

inclusion mask7 alternating prefix sum := sum(3, 2)

inclusion mask7 alternating prefix sum = 5

next NONEMPTY NUMERIC bitmask=7;ORIGINAL family indices=[0, 1, 2];actual selected original sets=[[23, 24, 25], [23, 25, 26], [23, 25, 27]]

FULL intersection members=[23, 25];size=2;ODD order sign=1;alternating prefix=5;zero-size terms retained

inclusion mask8 signed intersection cardinality := mul(1, 2)

inclusion mask8 signed intersection cardinality = 2

inclusion mask8 alternating prefix sum := sum(5, 2)

inclusion mask8 alternating prefix sum = 7

next NONEMPTY NUMERIC bitmask=8;ORIGINAL family indices=[3];actual selected original sets=[[23, 28]]

FULL intersection members=[23, 28];size=2;ODD order sign=1;alternating prefix=7;zero-size terms retained

inclusion mask9 signed intersection cardinality := mul(-1, 1)

inclusion mask9 signed intersection cardinality = -1

inclusion mask9 alternating prefix sum := sum(7, -1)

inclusion mask9 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=9;ORIGINAL family indices=[0, 3];actual selected original sets=[[23, 24, 25], [23, 28]]

FULL intersection members=[23];size=1;EVEN order sign=-1;alternating prefix=6;zero-size terms retained

inclusion mask10 signed intersection cardinality := mul(-1, 1)

inclusion mask10 signed intersection cardinality = -1

inclusion mask10 alternating prefix sum := sum(6, -1)

inclusion mask10 alternating prefix sum = 5

next NONEMPTY NUMERIC bitmask=10;ORIGINAL family indices=[1, 3];actual selected original sets=[[23, 25, 26], [23, 28]]

FULL intersection members=[23];size=1;EVEN order sign=-1;alternating prefix=5;zero-size terms retained

inclusion mask11 signed intersection cardinality := mul(1, 1)

inclusion mask11 signed intersection cardinality = 1

inclusion mask11 alternating prefix sum := sum(5, 1)

inclusion mask11 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=11;ORIGINAL family indices=[0, 1, 3];actual selected original sets=[[23, 24, 25], [23, 25, 26], [23, 28]]

FULL intersection members=[23];size=1;ODD order sign=1;alternating prefix=6;zero-size terms retained

inclusion mask12 signed intersection cardinality := mul(-1, 1)

inclusion mask12 signed intersection cardinality = -1

inclusion mask12 alternating prefix sum := sum(6, -1)

inclusion mask12 alternating prefix sum = 5

next NONEMPTY NUMERIC bitmask=12;ORIGINAL family indices=[2, 3];actual selected original sets=[[23, 25, 27], [23, 28]]

FULL intersection members=[23];size=1;EVEN order sign=-1;alternating prefix=5;zero-size terms retained

inclusion mask13 signed intersection cardinality := mul(1, 1)

inclusion mask13 signed intersection cardinality = 1

inclusion mask13 alternating prefix sum := sum(5, 1)

inclusion mask13 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=13;ORIGINAL family indices=[0, 2, 3];actual selected original sets=[[23, 24, 25], [23, 25, 27], [23, 28]]

FULL intersection members=[23];size=1;ODD order sign=1;alternating prefix=6;zero-size terms retained

inclusion mask14 signed intersection cardinality := mul(1, 1)

inclusion mask14 signed intersection cardinality = 1

inclusion mask14 alternating prefix sum := sum(6, 1)

inclusion mask14 alternating prefix sum = 7

next NONEMPTY NUMERIC bitmask=14;ORIGINAL family indices=[1, 2, 3];actual selected original sets=[[23, 25, 26], [23, 25, 27], [23, 28]]

FULL intersection members=[23];size=1;ODD order sign=1;alternating prefix=7;zero-size terms retained

inclusion mask15 signed intersection cardinality := mul(-1, 1)

inclusion mask15 signed intersection cardinality = -1

inclusion mask15 alternating prefix sum := sum(7, -1)

inclusion mask15 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=15;ORIGINAL family indices=[0, 1, 2, 3];actual selected original sets=[[23, 24, 25], [23, 25, 26], [23, 25, 27], [23, 28]]

FULL intersection members=[23];size=1;EVEN order sign=-1;alternating prefix=6;zero-size terms retained

inclusion COMPLETE actual union cardinality := sum(1, 1, 1, 1, 1, 1)

inclusion COMPLETE actual union cardinality = 6

ACTUAL union element=23;ALL containing ORIGINAL family indices=[0, 1, 2, 3];indexed multiplicity r=4

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

ACTUAL union element=24;ALL containing ORIGINAL family indices=[0];indexed multiplicity r=1

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

ACTUAL union element=25;ALL containing ORIGINAL family indices=[0, 1, 2];indexed multiplicity r=3

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

ACTUAL union element=26;ALL containing ORIGINAL family indices=[1];indexed multiplicity r=1

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

ACTUAL union element=27;ALL containing ORIGINAL family indices=[2];indexed multiplicity r=1

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

ACTUAL union element=28;ALL containing ORIGINAL family indices=[3];indexed multiplicity r=1

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

COMPLETE indexed family size=4;ALL15 nonempty index subsets accounted for;ACTUAL union=[23, 24, 25, 26, 27, 28]

each covered element has coefficient1 and any uncovered element coefficient0;FULL alternating sum=6=actual union size=6;empty family yields empty union/no terms/0

Reverse checks

• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S048 · premise_and_result_audit — Recompute inclusion COMPLETE actual union cardinality with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Recompute inclusion mask15 alternating prefix sum with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute inclusion mask15 signed intersection cardinality with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Recompute inclusion mask14 alternating prefix sum with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute inclusion mask14 signed intersection cardinality with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S040 · premise_and_result_audit — Recompute inclusion mask13 alternating prefix sum with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute inclusion mask13 signed intersection cardinality with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Recompute inclusion mask12 alternating prefix sum with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute inclusion mask12 signed intersection cardinality with exact rational arithmetic, not rounded decimals.
• 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 mask11 alternating prefix sum with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute inclusion mask11 signed intersection cardinality with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Recompute inclusion mask10 alternating prefix sum with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute inclusion mask10 signed intersection cardinality with exact rational arithmetic, not rounded decimals.
• 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 — Recompute inclusion mask9 alternating prefix sum with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute inclusion mask9 signed intersection cardinality with exact rational arithmetic, not rounded decimals.
• 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 — Recompute inclusion mask8 alternating prefix sum with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute inclusion mask8 signed intersection 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

(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)

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.