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Question 5389: Overlap-Corrected Finite Choice Counts

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0009-05389 · q-20261008-advance-20261012-r02-counting-principle-0089

Original question and givens

Finite integer choice sets A=[0, 1, 2, 3], B=[0, 1, 2, 3]. Give (|A|+|B|, |A∪B|, |A∩B|).

  • G001 — Finite integer choice sets A=[0, 1, 2, 3], B=[0, 1, 2, 3]. Give (|A|+|B|, |A∪B|, |A∩B|).

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.

Step-by-step derivation

S001 · implies

Before

declared finite tagged choices or full-product protocol

After

combinatorial counts,not probabilities or independence

Conditions: none

Addition uses disjoint tags and multiplication uses the declared full cross-stage protocol;neither is a probability-independence assumption.

Step identifiers and dependencies

Rule: finite_counting_protocol_scope · Uses: G001

S002 · implies

Before

choice=0;in A=Yes;in B=Yes

After

shared=Yes

Conditions: none

A shared element appears twice in the sum of sizes but once in the union.

Step identifiers and dependencies

Rule: counting_overlap_membership · Uses: G001, S001

S003 · implies

Before

choice=1;in A=Yes;in B=Yes

After

shared=Yes

Conditions: none

A shared element appears twice in the sum of sizes but once in the union.

Step identifiers and dependencies

Rule: counting_overlap_membership · Uses: G001, S002

S004 · implies

Before

choice=2;in A=Yes;in B=Yes

After

shared=Yes

Conditions: none

A shared element appears twice in the sum of sizes but once in the union.

Step identifiers and dependencies

Rule: counting_overlap_membership · Uses: G001, S003

S005 · implies

Before

choice=3;in A=Yes;in B=Yes

After

shared=Yes

Conditions: none

A shared element appears twice in the sum of sizes but once in the union.

Step identifiers and dependencies

Rule: counting_overlap_membership · Uses: G001, S004

S006 · calculation

Before

sum of choice-set sizes := sum(4, 4)

After

sum of choice-set sizes = 8

Conditions: none

First add the two set sizes,which may double-count shared choices.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

overlap-corrected union size := sub(8, 4)

After

overlap-corrected union size = 4

Conditions: none

Subtract the intersection once and audit the actual union size.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

A∪B=[0, 1, 2, 3];A∩B=[0, 1, 2, 3]

After

union size=4

Conditions: none

Without disjoint tags,apply the overlap correction rather than treating one shared choice as two outcomes.

Step identifiers and dependencies

Rule: finite_choice_inclusion_exclusion · Uses: G001, S007

Reverse checks

  • 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 overlap-corrected union size with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute sum of choice-set sizes 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

(8, 4, 4)

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: c2e09497de73111568441455b6bec91677fcae1d84da3f51f40dc0d5ee0baacc
  • candidate record SHA-256: fc03c0b0fe84975f9dbb368de87f628d93a17b4c9f54acff15dc026b63b47aae
  • semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
  • question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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