Let's go through each statement in the image and analyze its validity.

Statements Analysis

1. Statement 1:
   "Если $f : X \to Y, h : Y \to Z$ — биективные отображения, то композиция $h \circ f$ также биективна."

   • Explanation: A composition of bijective functions is also bijective. Since both $f$ and $h$ are bijections, their composition $h \circ f$ will also be a bijection.
   • Conclusion: This statement is true.

2. Statement 2:
   "Пусть $T$ — множество треугольников на плоскости (равные треугольники не различаются), $\mathbb{R}$ — множество действительных чисел, $f : T \to \mathbb{R}$ — отображение, заданное формулой $f(\Delta) = P_\Delta$ — периметр треугольника $\Delta$. Данное отображение не сюръективно и не инъективно."

   • Explanation:
     • The function $f$ maps triangles to their perimeters. This mapping is not injective because different triangles (e.g., triangles with different shapes but the same perimeter) could map to the same perimeter.
     • Additionally, it is not surjective because not every real number can be the perimeter of a triangle (e.g., negative numbers or extremely large values).
   • Conclusion: This statement is true.

3. Statement 3:
   "На множестве натуральных чисел $\mathbb{N}$ отображение $f(x) = x^2$ сюръективно."

   • Explanation:
     • The function $f(x) = x^2$ is not surjective on $\mathbb{N}$ (the natural numbers) because it only covers perfect squares and does not map to every natural number.
   • Conclusion: This statement is false.

4. Statement 4:
   "Если отображение $f : X \to Y$ сюръективно, то всегда существует обратное отображение $f^{-1}$."

   • Explanation:
     • Surjectivity alone is not sufficient for a function to have an inverse. For a function to be invertible, it must be bijective (both injective and surjective). A surjective function without injectivity cannot have an inverse.
   • Conclusion: This statement is false.

Summary of Correct Statements

The correct statements are:

• Statement 1: The composition of bijections is bijective.
• Statement 2: The function mapping triangles to their perimeters is neither injective nor surjective.

Would you like more details on any part of this explanation, or do you have further questions?

Additional Related Questions

1. What is the definition of a bijective function?
2. Can a function be invertible if it is only injective?
3. Why is the function $f(x) = x^2$ not surjective over the set of all integers?
4. How does injectivity and surjectivity affect the invertibility of a function?
5. Are there other geometric properties of triangles that could produce an injective function?

Tip

When analyzing the invertibility of a function, always check both injectivity and surjectivity conditions, as both are required for a function to have an inverse.