Problemas matemáticos y razonamiento · Set Overlap & Inclusion-Exclusion
Visualizador de eslabones de cadena e inclusión-exclusión
Visualizador de longitud de cadenas con eslabones entrelazados y principio de inclusión-exclusión
- Cálculo en cliente
- Sin registro
- Derivación en vivo
- Modo proyector
- Hoja de trabajo A4
Ventana de demostración matemática● 60 FPS
Variables de control del experimento
cm
cm
Derivaciones y métricas en tiempo real
Longitud total estirada
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Longitud consumida por solapamiento
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Uniones totales
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Longitud interior libre
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Derivación de fórmulas en tiempo real
Fórmula básica:
--Sustitución de valores:
--Conclusión: --
Ejemplos clásicos y análisis de errores en exámenes
Worked Problem
Problema: 10 circular iron links (outer diameter 8cm, wall thickness 0.5cm) interlock into a straight chain. Find total length.
Derivación paso a paso:
- 9 joints. Each joint consumes 2 * 0.5cm = 1.0cm. Total length = 10*8 - 9*1 = 71cm.
Worked Problem
Problema: Class of 45: 28 join math, 25 join English, 12 join both. How many join neither?
Derivación paso a paso:
- At least one: |A ∪ B| = 28 + 25 - 12 = 41. Neither: 45 - 41 = 4 students.
Hoja de indagación en clase y tareas
Q1. Variant 1: Identical links (outer 12cm, wall 1cm) chain to total 92cm. How many links are there?
Answer Key: First link gives 12cm; each subsequent link adds inner diameter 10cm. 12 + 10*(N-1) = 92 => N-1 = 8 => N = 9 links.
Q2. Variant 2: 8 paper strips (each 20cm long) are glued with 3cm overlap joints. Find total assembled length.
Answer Key: 7 overlap joints. Total length = 8*20 - 7*3 = 160 - 21 = 139cm.
Q3. Variant 3: Three 80cm poles are spliced with 15cm overlap joints. Find total spliced length.
Answer Key: 2 joints. Total length = 3*80 - 2*15 = 240 - 30 = 210cm.
Preguntas frecuentes y errores conceptuales
Why does each chain joint consume TWO wall thicknesses instead of one?
Because the right wall of the first link and left wall of the second link BOTH occupy the overlap zone! Neglecting the 2nd thickness is the classic error.
Why do closed loops have N joints while straight chains have N-1?
The closed-perimeter tree problem! In an open line, N items have N-1 gaps. When looped closed, the last link loops into the first, creating N joints.
What does Inclusion-Exclusion mean? What is the 3-set Venn formula?
Add individual sizes, subtract pairwise overlaps, and add back the 3-way intersection: |A∪B∪C| = A + B + C - (AB+BC+CA) + ABC.
When two overlapping rectangles lie on a table, what is the net covered area?
Total covered area = Area(A) + Area(B) - Overlap Area. Subtracting overlap removes the double-counted duplicate layer.