The image contains two exercises, labeled as Exercice 15 and Exercice 16. Here's a breakdown of both exercises:

Exercice 15:

This exercise discusses a function $f$ and asks to show that the function $f$ is bijective under certain conditions. The exact statement is partially cut off, but it mentions something like:

• "Soit $f$ une application d'un ensemble non vide $E$ dans lui-même telle que $f \circ f = f$. Montrer que $f$ est bijective."

This means:

• Let $f$ be a function from a non-empty set $E$ to itself, such that $f \circ f = f$. Prove that $f$ is bijective.

This problem likely involves proving that the function is both injective (one-to-one) and surjective (onto), given the idempotent property $f \circ f = f$.

Exercice 16:

This exercise is about characteristic functions and involves the following:

• Let $X$ be a set. If $A \subset X$, denote by $1_A$ the characteristic function of $A$.
• The function $\Psi \colon P(E) \to F(E, \{0, 1\})$ is defined by $A \mapsto 1_A$, where $P(E)$ represents the power set of $E$, and $F(E, \{0, 1\})$ denotes the set of functions from $E$ to $\{0, 1\}$.
• The task is to show something specific about this function $\Psi$, though the exact proof required is partially cut off in the image.

Both exercises deal with fundamental concepts in set theory and functions:

• Exercice 15 likely focuses on demonstrating bijectivity based on the idempotent nature of the function.
• Exercice 16 involves characteristic functions and the relation between the power set and functions mapping to $\{0,1\}$.

Do you want details on how to prove these exercises or further explanation of the concepts?

5 Related Questions:

1. What does it mean for a function to be bijective?
2. How do we prove injectivity and surjectivity separately?
3. What is the characteristic function of a set?
4. How does the function composition $f \circ f = f$ affect the properties of $f$?
5. What is the power set of a given set, and how is it related to characteristic functions?

Tip:

When proving bijectivity, first check if the function is injective by assuming two elements map to the same image and showing that the elements must be equal. Then, prove surjectivity by showing that for every element in the codomain, there is a preimage in the domain.