https://file.aiquickdraw.com/mathbot/file/17273566672084fz0qu8s.jpg

The image presents a list of mathematical statements in Russian. Let's analyze each statement to determine which ones are true.

1. **Множество всех преобразований множества \( M \) с введённой операцией композиции преобразований является полугруппой с единицей.**
   - Translation: "The set of all transformations of a set \( M \) with the introduced operation of composition of transformations is a semigroup with an identity element."
   - **True**: The set of transformations (functions) under composition generally forms a semigroup, and if there is an identity transformation (which leaves all elements unchanged), it also has an identity element.

2. **На множестве натуральных чисел у каждого элемента есть обратный.**
   - Translation: "In the set of natural numbers, every element has an inverse."
   - **False**: The natural numbers \( \mathbb{N} \) do not have inverses under addition or multiplication. For instance, under addition, there is no natural number \( x \) such that \( x + 1 = 0 \) (since 0 is not a natural number), and similarly for multiplication, no inverse exists for most elements.

3. **Инъективный гомоморфизм это изоморфизм.**
   - Translation: "An injective homomorphism is an isomorphism."
   - **False**: An injective homomorphism is not necessarily an isomorphism. For it to be an isomorphism, it must also be surjective (bijective). Injectivity alone only implies that it’s one-to-one, but not necessarily onto.

4. **Любая полугруппа изоморфна самой себе.**
   - Translation: "Any semigroup is isomorphic to itself."
   - **True**: This is a tautology. Every algebraic structure, including semigroups, is trivially isomorphic to itself via the identity map.

### Correct Answers:
- The first and fourth statements are **true**.

Would you like any more details on these concepts, or further elaboration?

Here are five related questions to deepen understanding:
1. What is the difference between a semigroup and a monoid?
2. Why don’t natural numbers have inverses in their set under common operations?
3. Can you provide an example of a non-surjective injective homomorphism?
4. What is the difference between injective, surjective, and bijective mappings?
5. Can semigroups exist without an identity element?

**Tip**: Always verify the properties of algebraic structures (like groups or semigroups) by checking both the operation and the identity or inverse properties.

Укажите все верные утверждения: Множество всех преобразований множества M с введённой операцией композиции преобразований является полугруппой с единицей. На множестве натуральных чисел у каждого элемента есть обратный. Инъективный гомоморфизм это изоморфизм. Любая полугруппа изоморфна самой себе.

This page analyzes a set of mathematical statements regarding semigroups, homomorphisms, and natural numbers. Topics include the identity element in semigroups, inverse elements in natural numbers, and isomorphisms. Suitable for university-level algebra students.

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