10 Conditions on Relations

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

• Section 10.1: Functional and Total Relations
  • Subsection 10.1.1: Functional Relations
    • Definition 10.1.1.1.1: Functional Relations
    • Proposition 10.1.1.1.2: Properties of Functional Relations
  • Subsection 10.1.2: Total Relations
    • Definition 10.1.2.1.1: Total Relations
    • Proposition 10.1.2.1.2: Properties of Total Relations
• Section 10.2: Reflexive Relations
  • Subsection 10.2.1: Foundations
    • Definition 10.2.1.1.1: Reflexive Relations
    • Remark 10.2.1.1.2: Unwinding Definition 10.2.1.1.1
    • Definition 10.2.1.1.3: The Po/Set of Reflexive Relations on a Set
    • Proposition 10.2.1.1.4: Properties of Reflexive Relations
  • Subsection 10.2.2: The Reflexive Closure of a Relation
    • Definition 10.2.2.1.1: The Reflexive Closure of a Relation
    • Construction 10.2.2.1.2: The Reflexive Closure of a Relation
    • Proposition 10.2.2.1.3: Properties of the Reflexive Closure of a Relation
• Section 10.3: Symmetric Relations
  • Subsection 10.3.1: Foundations
    • Definition 10.3.1.1.1: Symmetric Relations
    • Remark 10.3.1.1.2: Unwinding Definition 10.3.1.1.1
    • Definition 10.3.1.1.3: The Po/Set of Symmetric Relations on a Set
    • Proposition 10.3.1.1.4: Properties of Symmetric Relations
  • Subsection 10.3.2: The Symmetric Closure of a Relation
    • Definition 10.3.2.1.1: The Symmetric Closure of a Relation
    • Construction 10.3.2.1.2: The Symmetric Closure of a Relation
    • Proposition 10.3.2.1.3: Properties of the Symmetric Closure of a Relation
• Section 10.4: Transitive Relations
  • Subsection 10.4.1: Foundations
    • Definition 10.4.1.1.1: Transitive Relations
    • Remark 10.4.1.1.2: Unwinding Definition 10.4.1.1.1
    • Definition 10.4.1.1.3: The Po/Set of Transitive Relations on a Set
    • Proposition 10.4.1.1.4: Properties of Transitive Relations
  • Subsection 10.4.2: The Transitive Closure of a Relation
    • Definition 10.4.2.1.1: The Transitive Closure of a Relation
    • Construction 10.4.2.1.2: The Transitive Closure of a Relation
    • Proposition 10.4.2.1.3: Properties of the Transitive Closure of a Relation
• Section 10.5: Equivalence Relations
  • Subsection 10.5.1: Foundations
    • Definition 10.5.1.1.1: Equivalence Relations
    • Example 10.5.1.1.2: The Kernel of a Function
    • Definition 10.5.1.1.3: The Po/Set of Equivalence Relations on a Set
  • Subsection 10.5.2: The Equivalence Closure of a Relation
    • Definition 10.5.2.1.1: The Equivalence Closure of a Relation
    • Construction 10.5.2.1.2: The Equivalence Closure of a Relation
    • Proposition 10.5.2.1.3: Properties of Equivalence Relations
• Section 10.6: Quotients by Equivalence Relations
  • Subsection 10.6.1: Equivalence Classes
    • Definition 10.6.1.1.1: Equivalence Classes
  • Subsection 10.6.2: Quotients of Sets by Equivalence Relations
    • Definition 10.6.2.1.1: Quotients of Sets by Equivalence Relations
    • Remark 10.6.2.1.2: Why Use “Equivalence” Relations for Quotient Sets
    • Proposition 10.6.2.1.3: Properties of Quotient Sets