Question 2762: Area of a Union of Two Half-Open Integer Rectangles

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 4) and B=[0, 3)×[0, 4) are half-open rectangles. Find area(A∪B) in squared length units.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. A=[0, 3)×[0, 4) and B=[0, 3)×[0, 4) 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

Step-by-step derivation

orthonormal equal-length coordinate units;half-open regions

area in squared length units

A width := sub(3, 0)

A width = 3

A height := sub(4, 0)

A height = 4

A area := mul(3, 4)

A area = 12

B width := sub(3, 0)

B width = 3

B height := sub(4, 0)

B height = 4

B area := mul(3, 4)

B area = 12

overlap raw width := sub(3, 0)

overlap raw width = 3

overlap raw height := sub(4, 0)

overlap raw height = 4

raw overlap widths=(3,4)

nonnegative overlap widths=(3,4)

intersection area := mul(3, 4)

intersection area = 12

sum of rectangle areas := sum(12, 12)

sum of rectangle areas = 24

union area := sub(24, 12)

union area = 12

A∪B=(A\B) disjoint-union B

area(A∪B)=12

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.