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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.

Sections

Equivalence Relation

An equivalence relation is a specific type of relation over a set that satisfies properties of reflexivity, symmetry, and transitivity.

21 Section Overview

Start current section content and materials

21.1 Definition of Equivalence Relation

This section introduces equivalence relations, detailing their properties—reflexivity, symmetry, and transitivity—and explains equivalence classes through examples involving modulo operations.

21.2 Example of an Equivalence Relation

This section introduces equivalence relations and their defining properties: reflexivity, symmetry, and transitivity.

21.3 Properties of Equivalence Relations

This section introduces equivalence relations, which must be reflexive, symmetric, and transitive, and explains equivalence classes with examples.

21.4 Equivalence Classes

This section introduces equivalence relations and equivalence classes, explaining their definitions and properties through examples, particularly in integer congruences.

21.5 Examples of Equivalence Classes

This section explores equivalence classes defined by equivalence relations, demonstrating their properties and providing practical examples using modulo arithmetic.

21.6 Disjoint Equivalence Classes

This section introduces the concept of equivalence relations and equivalence classes, emphasizing their properties and significance in set theory.

21.7 Implication of Equivalence Classes

This section introduces equivalence relations and equivalence classes, highlighting their definitions and properties.

21.8 Theorem on Equivalence Classes

This section introduces equivalence relations and classes, explaining their properties and significance in mathematics using integer modulo operations.

Learning Objectives

  • 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.

Practice Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting
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