Math Problem Statement

Укажите все верные утверждения

Solution

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 ll is parallel to line mm, then mm is parallel to ll (true).
      • Transitive: If ll is parallel to mm and mm is parallel to nn, then ll is parallel to nn (true).
    • Since parallelism meets these criteria, this statement is true.
  2. "Пусть XX — множество натуральных чисел, m,nXm, n \in X, тогда отношение mnm \sim n тогда и только тогда, когда mm делит nn. Это отношение является отношением эквивалентности."

    • This statement proposes a relation on natural numbers where mnm \sim n if mm divides nn.
    • For this to be an equivalence relation, it must also satisfy reflexivity, symmetry, and transitivity:
      • Reflexive: mm divides itself (true).
      • Symmetric: If mm divides nn, it does not imply that nn divides mm (false).
      • Since symmetry is not satisfied, this is not an equivalence relation.
    • Therefore, this statement is false.
  3. "Пусть XX — множество людей на планете, m,nXm, n \in X, тогда отношение mnm \sim n тогда и только тогда, когда mm и nn знакомы. Это отношение является отношением эквивалентности."

    • Here, mnm \sim n if two people mm and nn know each other.
    • For this to be an equivalence relation:
      • Reflexive: A person knows themselves (true).
      • Symmetric: If mm knows nn, then nn knows mm (true).
      • Transitive: If mm knows nn and nn knows pp, it does not imply that mm knows pp (false).
    • Since transitivity is not satisfied, this is not an equivalence relation.
    • Therefore, this statement is false.
  4. "Если на множестве XX задано отношение эквивалентности, то оно порождает разбиение этого множества на классы эквивалентности."

    • This statement claims that any equivalence relation on a set XX creates a partition of XX 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?


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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Math Problem Analysis

Mathematical Concepts

Equivalence Relations
Properties of Relations
Partition of Sets

Formulas

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Theorems

Equivalence Relation Theorem
Partition Theorem

Suitable Grade Level

Undergraduate