Question 31980: Relative Complement in an Explicit Universe

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0062-31980 · q-20260915-finite-relative-complement-0005

Original question and givens

Let the explicit universe be U={941, 942, 943, 944, 945} and A={941, 943}⊆U. Find the relative complement U\A, listing each element once in increasing order.

G001 — Let the explicit universe be U={941, 942, 943, 944, 945} and A={941, 943}⊆U. Find the relative complement U\A, listing each element once in increasing order.

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-set-operations

Step-by-step derivation

S001 · implies

Before

source roles = [qf-finite-relative-complement, no-domain-field, no-operation-field, no-form-field]

After

derived = [[numeric_sorted_set_STRING_or_empty_symbol]]

Conditions: none

Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction.

Step identifiers and dependencies

Rule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001

S002 · implies

Before

source roles = [U ORIGINAL explicit universe, [941, 942, 943, 944, 945]]

After

derived = [[941, 942, 943, 944, 945]]

Conditions: none

The BASE finite set is a DISTINCT integer list;do not silently repair duplicates.Sort numerically ONLY for derived display,retaining original data.Do not import the expectation100-element or SAFE bounds.

Step identifiers and dependencies

Rule: BASE_FINITE_set_distinct_INTEGER_list_no_silent_duplicate_fix · Uses: G001, S001

S003 · implies

Before

source roles = [A ORIGINAL subset, [941, 943]]

After

derived = [[941, 943]]

Conditions: none

Step identifiers and dependencies

Rule: BASE_FINITE_set_distinct_INTEGER_list_no_silent_duplicate_fix · Uses: G001, S002

S004 · implies

Before

source roles = [0, 941, [941, 943]]

After

derived = [No]

Conditions: none

For EACH ORIGINAL label of the EXPLICIT universe U,retain it ONLY when not in A.This is U\A,not A\U or complement in an unstated universe.

Step identifiers and dependencies

Rule: Complement_EACH_ORIGINAL_universe_label_NOT_in_A · Uses: G001, S003

S005 · implies

Before

source roles = [1, 942, [941, 943]]

After

derived = [Yes]

Conditions: none

For EACH ORIGINAL label of the EXPLICIT universe U,retain it ONLY when not in A.This is U\A,not A\U or complement in an unstated universe.

Step identifiers and dependencies

Rule: Complement_EACH_ORIGINAL_universe_label_NOT_in_A · Uses: G001, S004

S006 · implies

Before

source roles = [2, 943, [941, 943]]

After

derived = [No]

Conditions: none

For EACH ORIGINAL label of the EXPLICIT universe U,retain it ONLY when not in A.This is U\A,not A\U or complement in an unstated universe.

Step identifiers and dependencies

Rule: Complement_EACH_ORIGINAL_universe_label_NOT_in_A · Uses: G001, S005

S007 · implies

Before

source roles = [3, 944, [941, 943]]

After

derived = [Yes]

Conditions: none

For EACH ORIGINAL label of the EXPLICIT universe U,retain it ONLY when not in A.This is U\A,not A\U or complement in an unstated universe.

Step identifiers and dependencies

Rule: Complement_EACH_ORIGINAL_universe_label_NOT_in_A · Uses: G001, S006

S008 · implies

Before

source roles = [4, 945, [941, 943]]

After

derived = [Yes]

Conditions: none

For EACH ORIGINAL label of the EXPLICIT universe U,retain it ONLY when not in A.This is U\A,not A\U or complement in an unstated universe.

Step identifiers and dependencies

Rule: Complement_EACH_ORIGINAL_universe_label_NOT_in_A · Uses: G001, S007

S009 · implies

Before

source roles = [[941, 942, 943, 944, 945], [941, 943]]

After

derived = [[942, 944, 945], {942, 944, 945}, Yes]

Conditions: none

AFTER A⊆U qualification,verify disjointness and union BACK to U,without lost/added labels.Display numeric increasing order once per element,or∅for empty.Keep ORIGINAL U/A order and data unchanged.

Step identifiers and dependencies

Rule: Complement_COMPLETE_disjoint_partition_of_fixed_U_numeric_set_display · Uses: G001, S008

Reverse checks

S009 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S008 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S006 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S005 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S004 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S003 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

{942, 944, 945}

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: 8068dcf75f417d7e745358c95d03a47c601136414293f66c41fb939898482d24
candidate record SHA-256: eefbc340465248d46ceae6a430be15d48cbd0edb8fcb1f2357e5dce1ef193132
semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd
question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f
Lean: NOT_RUN · human review: NOT_PERFORMED

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