Math Problem Statement
Solution
Let's analyze each statement to determine whether it is correct.
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"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 is parallel to line , then is parallel to (true).
- Transitive: If is parallel to and is parallel to , then is parallel to (true).
- Since parallelism meets these criteria, this statement is true.
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"Пусть — множество натуральных чисел, , тогда отношение тогда и только тогда, когда делит . Это отношение является отношением эквивалентности."
- This statement proposes a relation on natural numbers where if divides .
- For this to be an equivalence relation, it must also satisfy reflexivity, symmetry, and transitivity:
- Reflexive: divides itself (true).
- Symmetric: If divides , it does not imply that divides (false).
- Since symmetry is not satisfied, this is not an equivalence relation.
- Therefore, this statement is false.
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"Пусть — множество людей на планете, , тогда отношение тогда и только тогда, когда и знакомы. Это отношение является отношением эквивалентности."
- Here, if two people and know each other.
- For this to be an equivalence relation:
- Reflexive: A person knows themselves (true).
- Symmetric: If knows , then knows (true).
- Transitive: If knows and knows , it does not imply that knows (false).
- Since transitivity is not satisfied, this is not an equivalence relation.
- Therefore, this statement is false.
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"Если на множестве задано отношение эквивалентности, то оно порождает разбиение этого множества на классы эквивалентности."
- This statement claims that any equivalence relation on a set creates a partition of 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:
- What are the formal definitions of reflexivity, symmetry, and transitivity?
- Can you give examples of relations that are not equivalence relations but satisfy some of the criteria?
- How does the concept of equivalence classes help in grouping objects in mathematics?
- Why does parallelism satisfy the conditions of an equivalence relation?
- 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
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