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.