# Question 31905: Cardinality of a Finite Event [Return to catalogue page 160](https://ocme.evemisslab.com/en/questions/page/160/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260918-finite-events/#q-20260918-finite-event-cardinality-0005) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0062-31905 · q-20260918-finite-event-cardinality-0005 ## Original question and givens Let S={17, 18, 19}, F=P(S), and E={18}. What is the cardinality of E? - G001 — Let S={17, 18, 19}, F=P(S), and E={18}. What is the cardinality of E? ## 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-event](https://ocme.evemisslab.com/en/mko/mko-event/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-finite-event-cardinality, no-domain-field, no-operation-field, no-form-field] After derived = [[cardinality_NUMBER_not_probability]] Conditions: none Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction. Step identifiers and dependenciesRule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001 S002 · impliesBefore source roles = [S original sample, [17, 18, 19]] After derived = [[17, 18, 19]] Conditions: none The BASE finite set is a DISTINCT integer list;do not silently repair duplicates.Sort numerically ONLY for derived display,retaining original data.Do not import the expectation100-element or SAFE bounds. Step identifiers and dependenciesRule: BASE_FINITE_set_distinct_INTEGER_list_no_silent_duplicate_fix · Uses: G001, S001 S003 · impliesBefore source roles = [E original event candidate, [18]] After derived = [[18]] Conditions: none The BASE finite set is a DISTINCT integer list;do not silently repair duplicates.Sort numerically ONLY for derived display,retaining original data.Do not import the expectation100-element or SAFE bounds. Step identifiers and dependenciesRule: BASE_FINITE_set_distinct_INTEGER_list_no_silent_duplicate_fix · Uses: G001, S002 S004 · impliesBefore source roles = [0, 18, [17, 18, 19]] After derived = [Yes] Conditions: none Check EACH candidate event element in ORIGINAL S.The empty event satisfies the subset condition vacuously.F=P(S)is not a probability law and does not imply equal likelihood. Step identifiers and dependenciesRule: Event_EACH_candidate_element_membership_in_ORIGINAL_S · Uses: G001, S003 S005 · impliesBefore source roles = [[17, 18, 19], [18], P(S)] After derived = [[Yes], [], Yes] Conditions: none With EXACT F=P(S),event qualification is precisely E⊆S.An outside E is a LEGAL No in subset-validity;do not delete the witness first.Cardinality,occurrence and probability are distinct questions. Step identifiers and dependenciesRule: Event_FULLpowerset_qualification_ALL_subset_flags_NOT_probability · Uses: G001, S004 S006 · calculationBefore Event cardinality ORIGINAL distinct element slot0 := sum(0, 1) After Event cardinality ORIGINAL distinct element slot0 = 1 Conditions: none Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [[17, 18, 19], [18]] After derived = [1] Conditions: none Count EACH distinct element for|E|.The answer is a NUMBER,not a fraction or probability.The empty event has cardinality0,even when S is empty.No uniform distribution is assumed. Step identifiers and dependenciesRule: Event_CARDINALITY_NUMBER_distinct_count_NO_probability_or_nonempty_S · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S006 · premise_and_result_audit — Recompute Event cardinality ORIGINAL distinct element slot0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S004 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S002 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate. ## Result 1 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: 049bf9c403474e4e2ed7498ffe9d6d26a0481159a40c8b22ffc75bc3d19446c0 - candidate record SHA-256: de761277a79f9ff75c8ea6ed7d1144e5944b6327fe8a97a416648f7a723c3d07 - semantic review receipt SHA-256: 963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd - question-first witness SHA-256: d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260918-finite-event-cardinality-0005.json)