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
    • Definition 7.1.1.1.1: Reflexive Relations
    • Remark 7.1.1.1.2: Unwinding Definition 7.1.1.1.1
    • Definition 7.1.1.1.3: The Po/Set of Reflexive Relations on a Set
    • Proposition 7.1.1.1.4: Properties of Reflexive Relations
  • 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
    • Definition 7.2.1.1.1: Symmetric Relations
    • Remark 7.2.1.1.2: Unwinding Definition 7.2.1.1.1
    • Definition 7.2.1.1.3: The Po/Set of Symmetric Relations on a Set
    • Proposition 7.2.1.1.4: Properties of Symmetric Relations
  • 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
    • Definition 7.3.1.1.1: Transitive Relations
    • Remark 7.3.1.1.2: Unwinding Definition 7.3.1.1.1
    • Definition 7.3.1.1.3: The Po/Set of Transitive Relations on a Set
    • Proposition 7.3.1.1.4: Properties of Transitive Relations
  • 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
    • Definition 7.5.1.1.1: 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