Chapter 6: Relations Diagraphs

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Chapter 6: Relations Diagraphs

6.1 Introduction to binary relations

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6.2 Properties of binary relations

Reflexive and anti-reflexive binary relations

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Symmetric binary relations

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Anti-symmetric binary relations

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6.2.5: Anti-symmetric binary relations.

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Transitive binary relations

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Proving and disproving properties of binary relations

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6.3 Directed graphs, paths, and cycles

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Example 6.3.1: The internet as a directed graph.

Pharmaceutical Company: D

C: Academy of Pediatrics

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Walks in directed graphs

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6.3.3: Walks in directed graphs.

Walk:

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Trails, circuits, paths, and cycles

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6.4 Composition of relations

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6.5 Graph powers and the transitive closure

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The transitive closure

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Example 6.5.1: Connectivity in sensor networks.

Induction overview for the proof of the Graph Power Theorem

Proof 6.5.1: Proof of the Graph Power Theorem.

Proof.

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6.6 Matrix multiplication and graph powers

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Check Show answer

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Theorem 6.6.2: Addition and graph union.

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Check Show answer

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Check Show answer

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6.7 Partial orders

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Upward drawings of partial orders

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Start

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6.8 Strict orders and directed acyclic graphs

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Fact 6.8.1: Strict orders are anti-symmetric.

Proof 6.8.1: Strict orders are anti-symmetric.

Proof.

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Directed acyclic graphs

Proof.

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Topological sorts of DAGs

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6.9 Equivalence relations

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Proof.

Proof.

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6.10 N-ary relations and relational databases

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The selection operation

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The projection operation

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