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R0056-28596 · q-20261004-advance-20261005-r05-random-variable-0046
The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 22), (3, -18)}. List the event {s∈S:X(s)≥-21} in increasing order. Return elementary outcomes, not a numerical probability.
G001 — The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 22), (3, -18)}. List the event {s∈S:X(s)≥-21} in increasing order. Return elementary outcomes, not a numerical probability.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.
Before
source roles = [qf-random-variable-threshold-event, rv_threshold]
After
derived = [inclusive_threshold_OUTCOME_set_not_probability]
Conditions: none
Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions.
Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001
Before
source roles = [[0, 1, 2, 3], powerset(S), [-20, -20, 22, -18]]
After
derived = [[[0, -20], [1, -20], [2, 22], [3, -18]]]
Conditions: none
Align each value with the ORIGINAL S slot,not independently sorted columns.The full powerset contains every preimage of this real-valued function,hence measurability.No invented uniform weights are needed for a value,range or event.
Rule: RV_full_powerset_measurability_ORIGINAL_outcome_value_alignment · Uses: G001, S001
Before
source roles = [[[0, -20], [1, -20], [2, 22], [3, -18]], -21]
After
derived = [[[0, -20, Yes], [1, -20, Yes], [2, 22, Yes], [3, -18, Yes]], [0, 1, 2, 3], {0, 1, 2, 3}]Conditions: none
Test X(s)>=t INCLUSIVELY at EVERY outcome,and collect the original s.Return the event set,not a value set or probability;no matches give the empty set.
Rule: RV_INCLUSIVE_threshold_ALL_outcome_preimages · Uses: G001, S002
S003 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.S002 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.{0, 1, 2, 3}The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 16e387cc56df6645cfae9d4e0a60234342ecd7743c92f5d9a98d838ef551df5acandidate record SHA-256: f5465ed21f55650f8233030d13aeb24fad8377f90b53dc19a1684f0922b8cd9fsemantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759fLean: NOT_RUN · human review: NOT_PERFORMED