CONTENTS Equivalence Classes Partition Intersection of Equivalence Relations Example Example Isomorphis

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1 Contents Chapter 1. Relations 8 1. Relations and Their Properties Definition of a Relation Directed Graphs Representing Relations with Matrices Example Inverse Relation Special Properties of Binary Relations Examples of Relations and their Properties Theorem 1.8.1: Connection Matrices v.s. Properties Combining Relations Example Definition of Composition Example Theorem : Characterization of Transitive Relations Connection Matrices v.s. Composition Closure of Relations Definition of the Closure of Relations Reflexive Closure Symmetric Closure Examples Relation Identities Characterization of Symmetric Relations Paths Paths v.s Composition Characterization of a Transitive Relation Connectivity Relation Characterization of the Transitive Closure Example Cycles Corollaries to Theorem Example Properties v.s Closure Equivalence Relations Definition of an Equivalence Relations Example 33 3

2 CONTENTS Equivalence Classes Partition Intersection of Equivalence Relations Example Example Isomorphism is an Equivalence Relation Equivalence Relation Generated by a Relation R Using Closures to find an Equivalence Relation Partial Orderings Definition of a Partial Order Examples Pseudo-Orderings Well-Ordered Relation Examples Lexicographic Order Examples and Strings Hasse or Poset Diagrams Example Maximal and Minimal Elements Least and Greatest Elements Upper and Lower Bounds Least Upper and Greatest Lower Bounds Lattices Example Topological Sorting Topological Sorting Algorithm Existence of a Minimal Element 54 Chapter 2. Graphs Introduction to Graphs and Graph Isomorphism The Graph Menagerie Representing Graphs and Graph Isomorphism Incidence Matrices Example Degree The Handshaking Theorem Example Theorem Handshaking Theorem for Directed Graphs Graph Invariants Example Proof of Section 1.10 Part 3 for simple graphs Connectivity 70

3 CONTENTS Connectivity Example Connectedness Examples Theorem Example Connected Component Example Cut Vertex and Edge Examples Counting Edges Connectedness in Directed Graphs Paths and Isomorphism Example Theorem Euler and Hamilton Paths Euler and Hamilton Paths Examples Necessary and Sufficient Conditions for an Euler Circuit Necessary and Sufficient Conditions for an Euler Path Hamilton Circuits Examples Sufficient Condition for a Hamilton Circuit Introduction to Trees Definition of a Tree Examples Roots Example Isomorphism of Directed Graphs Isomorphism of Rooted Trees Terminology for Rooted Trees m-ary Tree Counting the Elements in a Tree Level Number of Leaves Characterizations of a Tree Spanning Trees Spanning Trees Example Example Existence Spanning Forest Distance Search and Decision Trees 101

4 CONTENTS Binary Tree Example Decision Tree Example Tree Traversal Ordered Trees Universal Address System Tree Traversal Preorder Traversal Inorder Traversal Postorder Traversal Infix Form 111 Chapter 3. Boolean Algebra Boolean Functions Boolean Functions Example Binary Operations Example Boolean Identities Dual Representing Boolean Functions Representing Boolean Functions Example Example Functionally Complete Example NAND and NOR Abstract Boolean Algebras Abstract Boolean Algebra Examples of Boolean Algebras Duality More Properties of a Boolean Algebra Proof of Idempotent Laws Proof of Dominance Laws Proof of Theorem Property Proof of DeMorgan s Law Isomorphism Atoms Theorem Theorem Basis Theorem Logic Gates 139

5 CONTENTS Logic Gates Example NOR and NAND gates Example Half Adder Full Adder Minimizing Circuits Minimizing Circuits Example Karnaugh Maps Two Variables Three Variables Four Variables Quine-McCluskey Method 151

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