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Questions and Answers
What is an equivalence class?
What is an equivalence class?
What is the relationship between equivalence classes and partitions?
What is the relationship between equivalence classes and partitions?
What is a partition of a set S?
What is a partition of a set S?
What is the characteristic of the partition induced by an equivalence relation R on S?
What is the characteristic of the partition induced by an equivalence relation R on S?
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What is the relationship between an element in S and an equivalence class?
What is the relationship between an element in S and an equivalence class?
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What can be used to define an equivalence relation on S?
What can be used to define an equivalence relation on S?
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What is the relationship between an equivalence relation and a partition of S?
What is the relationship between an equivalence relation and a partition of S?
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Study Notes
Equivalence Relations
Equivalence Classes
- An equivalence class is a subset of a set S, where all elements are related to each other by an equivalence relation R.
- If a, b ∈ S, then [a] = [b] if and only if aRb (a is related to b).
- Equivalence classes are also called equivalence sets or classes of equivalence.
- If R is an equivalence relation on S, then the set of all equivalence classes is a partition of S.
- Each element in S belongs to exactly one equivalence class.
Partitioning Sets
- A partition of a set S is a collection of non-empty, disjoint subsets of S, whose union is S.
- An equivalence relation R on S induces a partition of S into equivalence classes.
- Each element in S belongs to exactly one cell of the partition.
- The partition induced by R is unique, and it is the finest partition of S that is compatible with R.
- If P is a partition of S, then there exists an equivalence relation R on S such that P is the partition induced by R.
Note: The concepts of equivalence classes and partitioning sets are closely related. An equivalence relation induces a partition of a set, and a partition of a set can be used to define an equivalence relation.
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Description
Learn about equivalence classes, partitioning sets, and how they are related. Understand the concepts of equivalence relations, disjoint subsets, and finest partitions.
