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The question asks about a school with 480 students, offering two exchange programs: one with an Italian school in Orbetello (during summer) and another with a French school in Amiens (during Easter). The analysis showed that 78 students participate in at least one program, and 1.25% of all students attend both. Participation in the French exchange is twice that of the Italian one.

The problem introduces sets:
- **Set \( I \)**: Students participating in the Italian exchange.
- **Set \( F \)**: Students in the French exchange.
- **Set \( G \)**: All students in the school.

The task asks:
- **a)** To represent the sets \( I \), \( F \), and \( G \) in a Venn diagram, including the cardinalities.
- **b)** To mark specific sets \( M_1 \) through \( M_4 \) and calculate their sizes, then describe them in context:
  - \( M_1 = I \cap F \) (students in both exchanges),
  - \( M_2 = I \setminus F \) (students in the Italian exchange only),
  - \( M_3 = F \setminus I \) (students in the French exchange only),
  - \( M_4 = F^c \cap I^c \) (students not in either exchange).

Let's break it down:

1. The total number of students is \( G = 480 \).
2. The number of students participating in both exchanges is \( 1.25\% \) of 480, i.e., \( I \cap F = 6 \).
3. There are 78 students in total in the exchanges, so \( |I \cup F| = 78 \).
4. Let \( |F| = x \) and \( |I| = \frac{x}{2} \) since the number of French exchange participants is twice that of the Italian exchange.
5. From the inclusion-exclusion principle:  
   \[
   |I \cup F| = |I| + |F| - |I \cap F| \quad \text{so} \quad 78 = \frac{x}{2} + x - 6
   \]

We can now calculate the sizes of \( I \), \( F \), and the subsets.

I'll proceed with the calculation.From the calculations:

- The number of students in the French exchange \( |F| \) is \( 56 \).
- The number of students in the Italian exchange \( |I| \) is \( 28 \) (since participation in the French exchange is twice that of the Italian one).
- The number of students not participating in any exchange is \( 402 \).

Now let's proceed to break down the sets for part **b)**:

- \( M_1 = I \cap F = 6 \) (students in both exchanges),
- \( M_2 = I \setminus F = 28 - 6 = 22 \) (students in only the Italian exchange),
- \( M_3 = F \setminus I = 56 - 6 = 50 \) (students in only the French exchange),
- \( M_4 = F^c \cap I^c = 402 \) (students in neither exchange).

Would you like a detailed diagram to represent this? Here are five relative questions for further exploration:

1. How does the inclusion-exclusion principle help in set theory problems?
2. Can you generalize the relationship between overlapping sets in other contexts?
3. What other applications does set theory have in real-world problems?
4. How would the results change if the percentage of overlap increased?
5. What is the impact of not accounting for overlap in statistical analyses?

**Tip:** Always ensure to account for overlaps in sets to avoid double-counting when using the inclusion-exclusion principle.

Eine weiterführende Schule bietet ihren 480 Schülern zwei Schüleraustauschprogramme an: eines mit einer italienischen Schule in Orbetello und eines mit einer französischen Schule in Amiens. Insgesamt nehmen 78 Schüler teil, und 1,25% aller Schüler besuchen beide Programme. Die Teilnahme am französischen Austausch ist doppelt so hoch wie beim italienischen Austausch. Die Aufgabe erfordert die Berechnung und Darstellung der Mengen I (italienischer Austausch), F (französischer Austausch) und G (Gesamtschülerzahl).

Solve a set theory problem involving students in two exchange programs at a school. Use the inclusion-exclusion principle to calculate the number of students in each program and those participating in both.

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