21. Equivalence Relation
The lecture introduces the concept of equivalence relations, which are defined by three main properties: reflexivity, symmetry, and transitivity. An example is given with integer congruences, showing how these properties apply. The discussion extends to equivalence classes, highlighting their formation and uniqueness, as well as the notable property that equivalence classes are either completely disjoint or identical.
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What we have learnt
- Equivalence relations require reflexivity, symmetry, and transitivity.
- An equivalence class is a subset containing all elements related to a particular element under an equivalence relation.
- Equivalence classes derived from any element are either identical or completely disjoint.
Key Concepts
- -- Equivalence Relation
- A relation that is reflexive, symmetric, and transitive.
- -- Equivalence Class
- The subset of a set formed by all elements that are equivalent to a specific element under an equivalence relation.
- -- Congruence Modulo
- A relationship between two integers where they yield the same remainder when divided by a modulus.
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