# Question 2763: Area of a Union of Two Half-Open Integer Rectangles [Return to catalogue page 14](https://ocme.evemisslab.com/en/questions/page/14/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261005-advance-20261006-r06-area-foundations/#q-20261005-advance-20261006-r06-area-0088) 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. R0004-02763 · q-20261005-advance-20261006-r06-area-0088 ## Original question and givens Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 9) and B=[1, 5)×[2, 5) are half-open rectangles. Find area(A∪B) in squared length units. - G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 0)×[0, 9) and B=[1, 5)×[2, 5) are half-open rectangles. Find area(A∪B) in squared length units. ## 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-area](https://ocme.evemisslab.com/en/mko/mko-area/) ## Step-by-step derivation S001 · impliesBefore orthonormal equal-length coordinate units;half-open regions After area in squared length units Conditions: none Axes are orthogonal with equal length units;half-open boundaries make adjacent unit cells disjoint.Area uses squared units,not length units. Step identifiers and dependenciesRule: half_open_area_convention · Uses: G001 S002 · calculationBefore A width := sub(0, 0) After A width = 0 Conditions: none Right minus left is the nonnegative width;translation of the origin contributes no area. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore A height := sub(9, 0) After A height = 9 Conditions: none Top minus bottom is the nonnegative height,using endpoints of the same rectangle. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore A area := mul(0, 9) After A area = 0 Conditions: none Rectangle area is width times height;a zero side makes the half-open rectangle empty,with area0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore B width := sub(5, 1) After B width = 4 Conditions: none Right minus left is the nonnegative width;translation of the origin contributes no area. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore B height := sub(5, 2) After B height = 3 Conditions: none Top minus bottom is the nonnegative height,using endpoints of the same rectangle. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore B area := mul(4, 3) After B area = 12 Conditions: none Rectangle area is width times height;a zero side makes the half-open rectangle empty,with area0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore overlap raw width := sub(0, 1) After overlap raw width = -1 Conditions: none Use the smaller right endpoint and larger left endpoint for the intersection. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore overlap raw height := sub(5, 2) After overlap raw height = 3 Conditions: none Use the inner vertical endpoints in the same way. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore raw overlap widths=(-1,3) After nonnegative overlap widths=(0,3) Conditions: none Disjoint or edge-touching rectangles have intersection area0;two negative raw widths must not create a fictitious positive area. Step identifiers and dependenciesRule: empty_intersection_clamp · Uses: G001, S009 S011 · calculationBefore intersection area := mul(0, 3) After intersection area = 0 Conditions: none Multiply the clamped nonnegative intersection lengths. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore sum of rectangle areas := sum(0, 12) After sum of rectangle areas = 12 Conditions: none First add the rectangle areas;the intersection is then counted twice. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore union area := sub(12, 0) After union area = 12 Conditions: none Finite inclusion-exclusion subtracts the intersection once,so each region is counted once. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore A∪B=(A\B) disjoint-union B After area(A∪B)=12 Conditions: none The rule covers containment,equal or empty rectangles,disjointness and overlap;plain addition is invalid when positive-area overlap exists. Step identifiers and dependenciesRule: rectangle_union_inclusion_exclusion · Uses: G001, S013 ## Reverse checks - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute union area with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute sum of rectangle areas with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute intersection area with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute overlap raw height with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute overlap raw width with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute B area with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute B height with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute B width with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute A area with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute A height with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute A width 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 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: be98cae3e1d3de8bc209119eae90952336659b8336745913390f1fdc4d8ce9b8 - candidate record SHA-256: 4d2c8f7a5a6d4ea9037de9a508f278756fff970ca8b7e1ff80b5b17f1928c56f - semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56 - question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r06-area-0088.json)