8 Equivalence Relations and Apartness Relations
This chapter contains some material about reflexive, symmetric, transitive, equivalence, and apartness relations.
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Section 8.1: Reflexive Relations
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Subsection 8.1.1: Foundations
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Subsection 8.1.2: The Reflexive Closure of a Relation
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Definition 8.1.2.1.1: The Reflexive Closure of a Relation
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Construction 8.1.2.1.2: The Reflexive Closure of a Relation
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Proposition 8.1.2.1.3: Properties of the Reflexive Closure of a Relation
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Section 8.2: Symmetric Relations
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Subsection 8.2.1: Foundations
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Subsection 8.2.2: The Symmetric Closure of a Relation
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Definition 8.2.2.1.1: The Symmetric Closure of a Relation
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Construction 8.2.2.1.2: The Symmetric Closure of a Relation
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Proposition 8.2.2.1.3: Properties of the Symmetric Closure of a Relation
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Section 8.3: Transitive Relations
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Subsection 8.3.1: Foundations
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Subsection 8.3.2: The Transitive Closure of a Relation
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Definition 8.3.2.1.1: The Transitive Closure of a Relation
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Construction 8.3.2.1.2: The Transitive Closure of a Relation
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Proposition 8.3.2.1.3: Properties of the Transitive Closure of a Relation
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Section 8.4: Equivalence Relations
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Subsection 8.4.1: Foundations
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Definition 8.4.1.1.1: Equivalence Relations
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Example 8.4.1.1.2: The Kernel of a Function
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Definition 8.4.1.1.3: The Po/Set of Equivalence Relations on a Set
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Subsection 8.4.2: The Equivalence Closure of a Relation
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Definition 8.4.2.1.1: The Equivalence Closure of a Relation
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Construction 8.4.2.1.2: The Equivalence Closure of a Relation
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Proposition 8.4.2.1.3: Properties of Equivalence Relations
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Section 8.5: Quotients by Equivalence Relations
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Subsection 8.5.1: Equivalence Classes
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Subsection 8.5.2: Quotients of Sets by Equivalence Relations
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Definition 8.5.2.1.1: Quotients of Sets by Equivalence Relations
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Remark 8.5.2.1.2: Why Use “Equivalence” Relations for Quotient Sets
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Proposition 8.5.2.1.3: Properties of Quotient Sets