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Question 34086: Minimum and Maximum of a Finite Real Set

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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R0067-34086 · q-20261005-advance-20261006-r01-order-relation-0036

Original question and givens

Under the usual real order, the explicit finite nonempty set has elements {-10, 0, 12}. Give (minimum, maximum).

  • G001 — Under the usual real order, the explicit finite nonempty set has elements {-10, 0, 12}. Give (minimum, maximum).

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

source roles = [qf-order-finite-extrema, [operation, values]]

After

derived = [finite_extrema, no-invented-domain]

Conditions: none

Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.

Step identifiers and dependencies

Rule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001

S002 · implies

Before

source roles = [[-10, 0, 12]]

After

derived = [[-10, 0, 12], [[-10, Yes], [0, Yes], [12, Yes]], -10, 12]

Conditions: none

This is a NONEMPTY finite SET of at most100 DISTINCT elements,not an observation multiset.Audit min≤x≤max for EVERY element and attainment at both endpoints.A singleton has equal extrema;the empty set is outside THIS protocol.

Step identifiers and dependencies

Rule: Order_FINITE_nonempty_DISTINCT_set_ALL_extrema_bounds · Uses: G001, S001

Reverse checks

  • S002 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
  • S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.

Result

(-10, 12)

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: 8e1ef0329a1355a46a9cabb15a89fccff42ab4066befd7cb12868eb0960a1a7e
  • candidate record SHA-256: 92b4dec2e175a0c690e56ec2a62bbe6d875da815573520863ca851c5e08450ba
  • semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
  • question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON