# Question 2790: Area of a Half-Open Unit-Cell Union [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-0065) 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. R0004-02790 · q-20261005-advance-20261006-r06-area-0065 ## Original question and givens Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units. - G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared 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 · impliesBefore [[-2, -2], [-1, -2], [0, -2], [1, -2], [2, -2], [-2, -1], [-1, -1], [0, -1], [1, -1], [2, -1], [-2, 0], [-1, 0], [0, 0], [1, 0]] After [[-2, -2], [-2, -1], [-2, 0], [-1, -2], [-1, -1], [-1, 0], [0, -2], [0, -1], [0, 0], [1, -2], [1, -1], [1, 0], [2, -2], [2, -1]] Conditions: none Each integer lower-left corner identifies one half-open unit cell;repeated listings of that cell count only once. Step identifiers and dependenciesRule: unit_cell_deduplication · Uses: G001, S001 S003 · impliesBefore [[-2, -2], [-2, -1], [-2, 0], [-1, -2], [-1, -1], [-1, 0], [0, -2], [0, -1], [0, 0], [1, -2], [1, -1], [1, 0], [2, -2], [2, -1]] After 14 distinct, pairwise disjoint unit cells Conditions: none Distinct integer half-open unit cells are disjoint and each has area1;finite additivity gives union area,and an empty union has area0. Step identifiers and dependenciesRule: disjoint_unit_cell_area · Uses: G001, S002 S004 · calculationBefore unit-cell union area := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) After unit-cell union area = 14 Conditions: none Add one square unit per distinct cell;the empty sum is0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 ## Reverse checks - S004 · premise_and_result_audit — Recompute unit-cell union area with exact rational arithmetic, not rounded decimals. - 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 14 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: a9a89da5268e5bb5a4be082c1d7667ae12553bbce8ba06d68f4e663e1f6f8ef5 - candidate record SHA-256: cff4be52c7073416c55b466aa66e09c6ad49fc0e6819591246d2139211109dfc - 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-0065.json)