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The image contains several mathematical exercises, primarily focused on set theory and cardinality. I will transcribe and explain each exercise for clarity:

### Exercice 7:
Let \( E \) be a set with \( n \) elements, and let \( A \subset E \) be a subset with \( p \) elements. What is the number of subsets of \( E \) that contain exactly one element from \( A \)?

### Exercice 8:
Let \( E \) be a finite set with cardinality \( n \), and \( A \subset E \) a subset of \( E \). Determine the number of disjoint subsets of \( E \) that are also subsets of \( A \).

### Exercice 9:
Let \( E \) be a finite set with cardinality \( n \), determine the cardinality of the set:
\[ P(E) \cap \{ X \subset Y \} \]

### Exercice 10:
Let \( E \) be a finite set with cardinality \( n \), determine the cardinality of the set:
\[ P(E)^2 : X \cap Y = \emptyset \]
This refers to pairs of subsets of \( E \) whose intersection is empty.

### Exercice 11:
Let \( E \) be a finite set with cardinality \( n \geq 1 \). Determine the number of subsets of \( E \) with even cardinality.

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Do you want detailed solutions for any of these exercises? Here are some follow-up questions that may help expand on this:

1. What are the general rules for counting subsets of a set with specific properties (e.g., disjoint subsets)?
2. How can the binomial coefficient \( \binom{n}{k} \) be used to solve Exercise 7?
3. What is the significance of even cardinality in Exercise 11?
4. How does the concept of the power set \( P(E) \) apply to Exercises 9 and 10?
5. How are disjoint sets defined and why is their intersection empty?

**Tip**: In problems involving subsets, particularly disjoint subsets or subsets with specific properties, understanding the structure of the power set is crucial.

The image contains several mathematical exercises focused on set theory and cardinality.

This set of exercises explores various topics in set theory, including the cardinality of finite sets, counting subsets, disjoint sets, and more. Solve problems related to binomial coefficients, the power set, and even subsets with detailed solutions. Ideal for university-level students studying discrete mathematics.

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