The uploaded image contains a series of mathematical exercises involving set theory and logical proofs. Here's an outline of what the exercises ask for:

Exercise Overview:

1. Exercice 1: Show the equality of two sets involving unions and intersections over the natural numbers.
2. Exercice 2: Compare power sets for union and intersection of two subsets $A$ and $B$.
3. Exercice 3: Prove logical equivalences and equalities involving three subsets $A$, $B$, and $C$.
4. Exercice 4: Work with the symmetric difference $A \Delta B$ and prove properties for subsets of $E$.
5. Exercice 5: Solve set equations $X \cup A = B$ and $X \cap A = B$ within the power set $\mathcal{P}(E)$.
6. Exercice 6: Prove that a specific sequence of sets $(B_1, B_2, \ldots, B_n)$ forms a partition of $E$.

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5 Related Questions:

1. How do you compute the power set $\mathcal{P}(E)$ of a given set $E$?
2. What is the definition of a partition of a set?
3. How can we interpret the symmetric difference $A \Delta B$ in terms of set operations?
4. What strategies are used to solve set equations?
5. How can unions and intersections over indexed families of sets be visualized?

Tip:

To simplify set operations, use Venn diagrams to visualize relationships between sets. This helps clarify equivalences and inclusions!