Mathematische Sachaufgaben & Denkmodelle · Set Overlap & Inclusion-Exclusion
Kettenglieder & Überlappungs-Visualisierer
Inklusion-Exklusion-Prinzip & Kettenüberlappung: Gesamtlänge bei Wandstärkenüberdeckung
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Experimentelle Steuerungsparameter
cm
cm
Echtzeit-Herleitung & Kennzahlen
Gesamtlänge gestreckt
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Durch Überlappung verbrauchte Länge
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Verbindungen gesamt
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Freie Innenlänge
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Echtzeit-Formelherleitung
Grundformel:
--Einsetzungsschritte:
--Ergebnis: --
Typische Beispielaufgaben & Prüfungsfallen
Worked Problem
Aufgabe: 10 circular iron links (outer diameter 8cm, wall thickness 0.5cm) interlock into a straight chain. Find total length.
Schrittweise Herleitung:
- 9 joints. Each joint consumes 2 * 0.5cm = 1.0cm. Total length = 10*8 - 9*1 = 71cm.
Worked Problem
Aufgabe: Class of 45: 28 join math, 25 join English, 12 join both. How many join neither?
Schrittweise Herleitung:
- At least one: |A ∪ B| = 28 + 25 - 12 = 41. Neither: 45 - 41 = 4 students.
Arbeitsblatt für den Unterricht & Hausaufgaben
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.
Häufige Fragen & typische Missverständnisse
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.