Question 32019: Directional Finite Set Difference

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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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R0063-32019 · q-20260915-finite-set-difference-0019

Original question and givens

Let A={781, 782, 784, 786} and B={782, 785, 786}. Find the directional difference A\B, listing each element once in increasing order.

G001 — Let A={781, 782, 784, 786} and B={782, 785, 786}. Find the directional difference A\B, 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-set-difference, no-required-operation, no-required-domain, []]

After

derived = [[numeric_sorted_set_STRING_or_empty]]

Conditions: none

BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.

Step identifiers and dependencies

Rule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001

S002 · implies

Before

source roles = [A ORIGINAL, [781, 782, 784, 786]]

After

derived = [[781, 782, 784, 786]]

Conditions: none

THIS BASE set bank requires distinct integer lists;do not silently repair duplicates.Sort ONLY derived display,retaining originals.Do not import another bank SAFE/100-element bound.

Step identifiers and dependencies

Rule: Sets_BASE_distinct_integer_lists_no_SAFE_or100cap_no_silent_repair · Uses: G001, S001

S003 · implies

Before

source roles = [B ORIGINAL, [782, 785, 786]]

After

derived = [[782, 785, 786]]

Conditions: none

THIS BASE set bank requires distinct integer lists;do not silently repair duplicates.Sort ONLY derived display,retaining originals.Do not import another bank SAFE/100-element bound.

Step identifiers and dependencies

Rule: Sets_BASE_distinct_integer_lists_no_SAFE_or100cap_no_silent_repair · Uses: G001, S002

S004 · implies

Before

source roles = [qf-finite-set-difference, 781, [781, 782, 784, 786], [782, 785, 786]]

After

derived = [Yes, No, Yes]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

Step identifiers and dependencies

Rule: Sets_EVERY_union_label_TWO_memberships_AND_directional_predicate · Uses: G001, S003

S005 · implies

Before

source roles = [qf-finite-set-difference, 782, [781, 782, 784, 786], [782, 785, 786]]

After

derived = [Yes, Yes, No]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

Step identifiers and dependencies

Rule: Sets_EVERY_union_label_TWO_memberships_AND_directional_predicate · Uses: G001, S004

S006 · implies

Before

source roles = [qf-finite-set-difference, 784, [781, 782, 784, 786], [782, 785, 786]]

After

derived = [Yes, No, Yes]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

Step identifiers and dependencies

Rule: Sets_EVERY_union_label_TWO_memberships_AND_directional_predicate · Uses: G001, S005

S007 · implies

Before

source roles = [qf-finite-set-difference, 785, [781, 782, 784, 786], [782, 785, 786]]

After

derived = [No, Yes, No]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

Step identifiers and dependencies

Rule: Sets_EVERY_union_label_TWO_memberships_AND_directional_predicate · Uses: G001, S006

S008 · implies

Before

source roles = [qf-finite-set-difference, 786, [781, 782, 784, 786], [782, 785, 786]]

After

derived = [Yes, Yes, No]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

Step identifiers and dependencies

Rule: Sets_EVERY_union_label_TWO_memberships_AND_directional_predicate · Uses: G001, S007

S009 · implies

Before

source roles = [qf-finite-set-difference, [781, 782, 784, 786], [782, 785, 786]]

After

derived = [[781, 784], {781, 784}, {785}]

Conditions: none

AFTER every label,display the complete numeric increasing set once per element,or∅.For difference ALSO compute B\A to distinguish direction.Keep original A/B and source hashes.

Step identifiers and dependencies

Rule: Sets_COMPLETE_extensional_result_numeric_order_reverse_difference_separate · Uses: G001, S008

Reverse checks

S009 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S008 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S007 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S006 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S005 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S004 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S003 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

{781, 784}

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: 4ba41c50267050d123bc39bd273b1690ba4d2872e3a3811840c08061ce4b41ce
candidate record SHA-256: a68f11da85f58ce0e408c28956c373a6e3759294383f80f2a23f2ea73ae531f1
semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0
question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88
Lean: NOT_RUN · human review: NOT_PERFORMED

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