22. Lecture -22
The lecture introduces the concept of equivalence relations and partitions, establishing a significant relationship between the two. It defines a partition as a collection of non-empty, pairwise disjoint subsets that form the original set when united. The discussion illustrates that equivalence classes resulting from an equivalence relation correspond to partitions of the set, indicating that the quantity of equivalence relations equals the number of partitions for a set.
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What we have learnt
- A partition of a set is a collection of non-empty, disjoint subsets whose union equals the original set.
- Equivalence classes formed by an equivalence relation create a partition of the set.
- The number of equivalence relations on a set is equal to the number of partitions of that set.
Key Concepts
- -- Partition
- A partition of a set is a collection of non-empty, pairwise disjoint subsets such that their union equals the original set.
- -- Equivalence Relation
- An equivalence relation on a set is a relation that is reflexive, symmetric, and transitive.
- -- Equivalence Class
- An equivalence class is a subset formed by grouping all elements that are equivalent to each other under a given equivalence relation.
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