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Question 6142: Ordered Opposites and Two Difference Directions

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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R0011-06142 · q-20261009-advance-20261013-r02-integer-0067

Original question and givens

Integers a=4, b=-15. Give (−a, −b, a−b, b−a), retaining signs and direction.

  • G001 — Integers a=4, b=-15. Give (−a, −b, a−b, b−a), retaining signs and direction.

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 exact integer or rational-representation protocol

After

retain signs,ordered operands and original index order

Conditions: none

Check these finite data with exact integer arithmetic,without floating approximations or a claim to prove all integer-ring axioms.

Step identifiers and dependencies

Rule: integer_arithmetic_scope · Uses: G001

S002 · calculation

Before

opposite of a := mul(-1, 4)

After

opposite of a = -4

Conditions: none

Multiply by−1 for the additive opposite;0 stays0.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

opposite of b := mul(-1, -15)

After

opposite of b = 15

Conditions: none

Negate b separately without changing its original value.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

a minus b := sub(4, -15)

After

a minus b = 19

Conditions: none

Compute the signed difference in the order a−b.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

b minus a := sub(-15, 4)

After

b minus a = -19

Conditions: none

Compute the reverse difference separately;it is the opposite of the first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

Reverse checks

  • S005 · premise_and_result_audit — Recompute b minus a with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute a minus b with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute opposite of b with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute opposite of a with exact rational arithmetic, not rounded decimals.
  • 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, 15, 19, -19)

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: 1e6528c66111d7d8cbdfc9e85743bc3079abf4786f1f2f414f7cab97088acc2c
  • candidate record SHA-256: 2cc858ad36dc7f2da444e44e3d1f827d7f8dc42d74ac96c2d6395a17a38fa191
  • semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
  • question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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