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Let's analyze each statement to determine whether it is correct.

1. **"Отношение параллельности двух прямых является отношением эквивалентности."**
   - This statement claims that the relationship of parallelism between two lines is an equivalence relation.
   - For an equivalence relation, it must satisfy reflexivity, symmetry, and transitivity:
     - Reflexive: A line is parallel to itself (true).
     - Symmetric: If line \( l \) is parallel to line \( m \), then \( m \) is parallel to \( l \) (true).
     - Transitive: If \( l \) is parallel to \( m \) and \( m \) is parallel to \( n \), then \( l \) is parallel to \( n \) (true).
   - Since parallelism meets these criteria, this statement is **true**.

2. **"Пусть \( X \) — множество натуральных чисел, \( m, n \in X \), тогда отношение \( m \sim n \) тогда и только тогда, когда \( m \) делит \( n \). Это отношение является отношением эквивалентности."**
   - This statement proposes a relation on natural numbers where \( m \sim n \) if \( m \) divides \( n \).
   - For this to be an equivalence relation, it must also satisfy reflexivity, symmetry, and transitivity:
     - Reflexive: \( m \) divides itself (true).
     - Symmetric: If \( m \) divides \( n \), it does not imply that \( n \) divides \( m \) (false).
     - Since symmetry is not satisfied, this is **not an equivalence relation**.
   - Therefore, this statement is **false**.

3. **"Пусть \( X \) — множество людей на планете, \( m, n \in X \), тогда отношение \( m \sim n \) тогда и только тогда, когда \( m \) и \( n \) знакомы. Это отношение является отношением эквивалентности."**
   - Here, \( m \sim n \) if two people \( m \) and \( n \) know each other.
   - For this to be an equivalence relation:
     - Reflexive: A person knows themselves (true).
     - Symmetric: If \( m \) knows \( n \), then \( n \) knows \( m \) (true).
     - Transitive: If \( m \) knows \( n \) and \( n \) knows \( p \), it does not imply that \( m \) knows \( p \) (false).
   - Since transitivity is not satisfied, this is **not an equivalence relation**.
   - Therefore, this statement is **false**.

4. **"Если на множестве \( X \) задано отношение эквивалентности, то оно порождает разбиение этого множества на классы эквивалентности."**
   - This statement claims that any equivalence relation on a set \( X \) creates a partition of \( X \) into equivalence classes.
   - This is a fundamental property of equivalence relations: they partition a set into mutually exclusive equivalence classes.
   - Thus, this statement is **true**.

### Conclusion
The correct statements are:
- **Statement 1**: Parallelism between lines is an equivalence relation.
- **Statement 4**: An equivalence relation partitions the set into equivalence classes.

Would you like further explanation on any of these points or have additional questions? 

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Here are five related questions to deepen understanding:
1. What are the formal definitions of reflexivity, symmetry, and transitivity?
2. Can you give examples of relations that are not equivalence relations but satisfy some of the criteria?
3. How does the concept of equivalence classes help in grouping objects in mathematics?
4. Why does parallelism satisfy the conditions of an equivalence relation?
5. Are there other real-life examples that naturally form equivalence relations?

**Tip:** To test if a relation is an equivalence relation, check reflexivity, symmetry, and transitivity in that order.

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Analyze statements about equivalence relations, such as parallelism and relationships on sets, to determine which are valid. This exercise covers core concepts of equivalence relations, including reflexivity, symmetry, and transitivity, and explains how equivalence classes partition sets.

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