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Question 4428: Sorted Symmetric Difference

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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R0007-04428 · q-20261008-advance-20261010-r02-set-operations-0028

Original question and givens

Finite integer sets A=[] and B=[-2, 2]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.

  • G001 — Finite integer sets A=[] and B=[-2, 2]; repeated elements count once. List (A\B)∪(B\A) in increasing order without repetition.

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

raw A=[];raw B=[-2, 2]

After

A=[];B=[-2, 2]

Conditions: none

Deduplicate and numerically sort integer sets;repetition and input order do not change membership.The original lists remain retained.

Step identifiers and dependencies

Rule: finite_set_extensionality · Uses: G001

S002 · implies

Before

element=-2

After

in A=No;in B=Yes

Conditions: none

Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.

Step identifiers and dependencies

Rule: finite_set_membership_row · Uses: G001, S001

S003 · implies

Before

element=2

After

in A=No;in B=Yes

Conditions: none

Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting.

Step identifiers and dependencies

Rule: finite_set_membership_row · Uses: G001, S002

S004 · implies

Before

A\B=[];B\A=[-2, 2]

After

[-2, 2]

Conditions: none

Retain elements belonging to exactly one side;shared elements must be excluded,unlike ordinary union.

Step identifiers and dependencies

Rule: symmetric_difference_exclusive_membership · Uses: G001, S003

Reverse checks

  • 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

[-2, 2]

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: f9b25ad1664f39392067181de6638e6da2585e955bbd3d4d35a343133af832d5
  • candidate record SHA-256: ecc8c3643cc7cdc8656b8dd2d4fae13a60fced15fbb19f5199c4fcabbdfe4aef
  • semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a
  • question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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