# Question 4500: Union and Intersection Cardinalities [Return to catalogue page 23](https://ocme.evemisslab.com/en/questions/page/23/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r02-set-operations-foundations/#q-20261008-advance-20261010-r02-set-operations-0025) 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. R0007-04500 · q-20261008-advance-20261010-r02-set-operations-0025 ## Original question and givens Finite integer sets A=[0, 1, 2] and B=[0, 1, 2, 100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|). - G001 — Finite integer sets A=[0, 1, 2] and B=[0, 1, 2, 100, 101]; repeated elements count once. Give (|A∪B|, |A∩B|). ## 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-set-operations](https://ocme.evemisslab.com/en/mko/mko-set-operations/) ## Step-by-step derivation S001 · impliesBefore raw A=[0, 1, 2];raw B=[0, 1, 2, 100, 101] After A=[0, 1, 2];B=[0, 1, 2, 100, 101] Conditions: none Deduplicate and numerically sort integer sets;repetition and input order do not change membership.The original lists remain retained. Step identifiers and dependenciesRule: finite_set_extensionality · Uses: G001 S002 · impliesBefore element=0 After in A=Yes;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 dependenciesRule: finite_set_membership_row · Uses: G001, S001 S003 · impliesBefore element=1 After in A=Yes;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 dependenciesRule: finite_set_membership_row · Uses: G001, S002 S004 · impliesBefore element=2 After in A=Yes;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 dependenciesRule: finite_set_membership_row · Uses: G001, S003 S005 · impliesBefore element=100 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 dependenciesRule: finite_set_membership_row · Uses: G001, S004 S006 · impliesBefore element=101 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 dependenciesRule: finite_set_membership_row · Uses: G001, S005 S007 · calculationBefore separate set cardinalities sum := sum(3, 5) After separate set cardinalities sum = 8 Conditions: none Adding the two set sizes counts their intersection twice. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore union cardinality check := sub(8, 3) After union cardinality check = 5 Conditions: none Subtract one intersection count and compare with the deduplicated union size. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore union=[0, 1, 2, 100, 101];intersection=[0, 1, 2] After (5, 3) Conditions: none Report union size before intersection size;the empty set has size0 and a shared element is counted once in the union. Step identifiers and dependenciesRule: union_intersection_cardinality · Uses: G001, S008 ## Reverse checks - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute union cardinality check with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute separate set cardinalities sum with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (5, 3) 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: be493093a7f939bd360b0227117e8a21dd219c638db11974be2943c677e3baef - candidate record SHA-256: f69efbc827ec8a599e71d654958ce14cec2f1db619116087b1e25f3e3df03553 - semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a - question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r02-set-operations-0025.json)