7 Equivalence Relations and Apartness Relations

This chapter contains some material about reflexive, symmetric, transitive, equivalence, and apartness relations.

  • Section 7.1: Reflexive Relations
    • Subsection 7.1.1: Foundations
    • Subsection 7.1.2: The Reflexive Closure of a Relation
      • Definition 7.1.2.1.1: The Reflexive Closure of a Relation
      • Construction 7.1.2.1.2: The Reflexive Closure of a Relation
      • Proposition 7.1.2.1.3: Properties of the Reflexive Closure of a Relation
  • Section 7.2: Symmetric Relations
    • Subsection 7.2.1: Foundations
    • Subsection 7.2.2: The Symmetric Closure of a Relation
      • Definition 7.2.2.1.1: The Symmetric Closure of a Relation
      • Construction 7.2.2.1.2: The Symmetric Closure of a Relation
      • Proposition 7.2.2.1.3: Properties of the Symmetric Closure of a Relation
  • Section 7.3: Transitive Relations
    • Subsection 7.3.1: Foundations
    • Subsection 7.3.2: The Transitive Closure of a Relation
      • Definition 7.3.2.1.1: The Transitive Closure of a Relation
      • Construction 7.3.2.1.2: The Transitive Closure of a Relation
      • Proposition 7.3.2.1.3: Properties of the Transitive Closure of a Relation
  • Section 7.4: Equivalence Relations
    • Subsection 7.4.1: Foundations
      • Definition 7.4.1.1.1: Equivalence Relations
      • Example 7.4.1.1.2: The Kernel of a Function
      • Definition 7.4.1.1.3: The Po/Set of Equivalence Relations on a Set
    • Subsection 7.4.2: The Equivalence Closure of a Relation
      • Definition 7.4.2.1.1: The Equivalence Closure of a Relation
      • Construction 7.4.2.1.2: The Equivalence Closure of a Relation
      • Proposition 7.4.2.1.3: Properties of Equivalence Relations
  • Section 7.5: Quotients by Equivalence Relations
    • Subsection 7.5.1: Equivalence Classes
    • Subsection 7.5.2: Quotients of Sets by Equivalence Relations
      • Definition 7.5.2.1.1: Quotients of Sets by Equivalence Relations
      • Remark 7.5.2.1.2: Why Use “Equivalence” Relations for Quotient Sets
      • Proposition 7.5.2.1.3: Properties of Quotient Sets

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