# Index Index

A Coset Counting Formula, Corollary

Abelian Group, Definition

a Limerick, Poem

Adjacency Matrix, Definition

Adjacency Matrix Method, Subsection

Algebraic Systems, Section

Alternating Group, Definition

Analog-to-digital Conversion, Example

Antisymmetric Relation, Definition

Associative Property, Definition

Atom of a Boolean Algebra, Definition

Automata, Section

Automorphism

Inner, Exercise

Basic Law Of Addition:, Theorem

Basic Set Operations, Section

Basis, Definition

Biconditional Proposition, Definition

Bijection, Definition

Binary Conversion Algorithm, Algorithm

Binary Operation., Definition

Binary Representation, Section

Binary Search, Subsection

Binary Tree, Definition

Binary Trees, Section

Binomial Coefficient, Definition

Recursive Definition, Definition

Binomial Coefficient Formula, Theorem

Binomial Theorem, The, Theorem

Bipartite Graph., Definition

Bipartitie

a Limerick, Poem

Boolean Algebra, Definition

Boolean Algebras, Section

Boolean Arithmetic, Definition

Boolean Expression, Definition

Boolean Expressions, Section

Bounded Lattice, Definition

Breadth-First Search, Subsection Algorithm

Bridge, Definition

Bubble Sort, Subsection

Cancellation in Groups, Theorem

Cardinality., Definition

Cartesian Product, Definition

Center of a Graph, Item

Characteristic Equation, Definition

Characteristic function, Exercise

Characteristic Roots, Definition

Chinese Remainder Theorem, Theorem

Chromatic Number, Definition

Closed Form Expression., Definition

Closest Neighbor Algorithm, Algorithm

Closure Property, Definition

Code

Polynomial, Example

Codes

Group, Section

Coding Theory, Section

Combinations, Subsection

Commutative Property, Definition

Complement of a Lattice Element, Definition

as an operation, Definition

Complement of a set, Definition

Complemented Lattice, Definition

Complete Undirected Graph., Definition

Composition of Functions, Definition

Composition of Relations, Definition

Concatenation, Definition

Conditional Statement, Definition

Congruence Modulo m, Definition

Conjunction, Logical, Definition

Connected Component, Definition

Connectivity in Graphs, Section

Contradiction, Definition

Contrapositive, Definition

Converse, Definition

Coset, Definition

Coset Representative, Definition

Cosets

Operation on, Definition

Cosets and Factor Groups, Section

Countable Set, Definition

Counting Binary Trees, Subsection

covering relation, Definition

CRT, Theorem

Cycle, Definition

Cycle Notation, Subsection

Cyclic Group, Definition Definition

Cyclic Subgroup, Definition

Degree, Definition

Degree Sequence of a Graph, Definition

Derangement, Subsection

Diagonalizable Matrix, Definition

Diagonalization Process, The, Section

Digraph, Paragraph

Dimension of a Vector Space, Definition

Direct Product, Definition

Direct Products, Section

Direct proof, Example

Directed graph, Paragraph Definition

Disjoint Cycles, Note

Disjoint Sets, Definition

Disjunction, Logical, Definition

Distributive Lattice, Definition

Distributive Property, Definition

Divides, Definition

Division Property for Integers, Theorem

Division Property for Polynomials, Theorem

Divisors of an Integer, Definition

Doyle, Chris, Poem

Duality for Boolean Algebras, Definition

Eigenvalue, Definition

Eigenvector, Definition

Elementary Operations on Equations, Theorem

Elementary Row Operations, Theorem

Embedding of a graph, Paragraph

Empty set, Paragraph

Equivalence, Definition

Equivalence Class, Item

Equivalence Relation, Definition

Equivalence Relations, Subsection

Euclidean Algorithm, The, Subsection

Euler's Formula, Theorem

Euler's Theorem, Theorem

Koenigsberg Case, Theorem

Eulerian Paths, Circuits, Graphs, Definition

Existential Quantifier, Definition

Exponentiation in Groups, Definition

Expression Tree, Subsection

Factor Group, Definition

Factor Theorem, Theorem

Factorial, Definition

Fibonacci Sequence, Example

Field, Definition

Finite-State Machine, Definition

Finite-State Machines, Section

Five-Color Theorem, Theorem

Flow Augmenting Path, Definition

Forest., Definition

Formal Language, Definition

Four-Color Theorem, Theorem

Free Monoids and Languages, Section

Function, Definition

Bijective, Definition

Composition, Definition

Equality, Definition

Injective, Definition

One-to-one, Definition

Onto, Definition

Surjective, Definition

Functions

Of two Variables, Subsection

Functions Between Two Sets

Set of, Definition

Fundamental Theorem of Group Homomorphisms, Theorem

Gauss-Jordan Algorithm, Algorithm

Generalized Set Operations, Definition

Generate, Definition

Generating Function, Definition

Generating Functions, Section

Closed form expressions for, Subsection

Operations on,, Subsection Definition

Generation Problem, Problem

Generator, Definition

George Boole

a Limerick, Poem

Graph

Data Structures, Section

Multigraph, Definition

Simple Directed, Definition

Undirected, Definition

Graph Coloring, Definition

Graph Optimization, Section

Graphic Sequence, Definition

Greatest Common Divisor (\(\gcd\)), Definition

Greatest Element, Definition

Greatest Lower Bound, Definition

Group, Definition

Hamiltonian Paths, Circuits, and Graphs, Definition

Hasse Diagram, Paragraph

Homogeneous Recurrence Relation., Definition

Homomorphism, Definition

Group, Section

Howlett,Chris, Poem

Idempotent Property, Definition

Identity Function, Definition

Identity Matrix, Definition

Identity Property, Definition

Image of an Element., Definition

Implication, Definition

Improper subset, Paragraph

Inclusion-Exclusion, Laws of, Theorem

Indirect proof, Paragraph

Induced Subgraph, Definition

Induction and Recursion, Subsection

Injection, Definition

Integers Modulo \(n\)

Additive Group, Definition

Multiplicative Group, Definition

Integral Domain, Definition

Intersection, Definition

Inverse

Matrix, Definition

Inverse Function

of a function on a set, Definition

Inverse Property, Definition

Involution Property, Definition

Irreducibility of a Polynomial, Definition

Isomorphic Graphs, Definition

Isomorphism

Group, Definition

Isomorphisms, Section

Join, Definition

Kernel, Definition

Kruskal's Algorithm, Algorithm

Lagrange's Theorem, Theorem

Lattice, Definition

Lattice Paths, Exercise

Lattices, Section

Laws of Matrix Algebra, Section

Leaf of a binary tree, Item

Leaf, of a binary tree, Item

Least Element, Definition

Least Upper Bound, Definition

Left Distributive Property, Definition

Level of a vertex, Paragraph

Levels of Abstraction, Subsection

Limerick

countably infinite, Poem

Enumerative Combinatorics, Poem

Linear Combination., Definition

Linear Dependence, Definition

Linear Equations

over the Integers Mod 2, Section

Linear Equations in a Group, Theorem

Linear Independence, Definition

Logarithm

General Base, Definition

Logarithm, base 2, Definition

Logarithms, Subsubsection

Properties, Theorem

Lower Bound, Definition

Machine of a Monoid, Section Definition

Many Faces of Recursion, The, Section

Matrix Addition, Definition

Matrix Inversion, Section

Matrix Multiplication, Definition

Matrix Oddities, Section

Maximal flow, Paragraph

Meet, Definition

Merge Sort, Subsection

Mesh Graph, Exercise

Minimal Spanning Tree, Definition

Minimum Diameter Spanning Tree, Definition

Minset, Definition

Minset Normal Form, Definition

Minterm, Definition

Minterm Normal Form, Definition

Modular Addition, Definition

Modular Arithmetic, Subsection

Properties, Subsection

Modular Multiplication, Definition

Monoid, Definition

of a Finite-State Machine, Section

Monoids, Section

Multigraph, Definition

Multiple Pop and Push:, Definition

Multiplicative Inverses, Definition

N-cube, Definition

Natural Homomorphism, Definition

Negation, Logical, Definition

Network, Definition

Networks, Subsection

Nonhomogeneous of Finite Order Linear Relations

Solution, Subsection

Normal Subgroup., Definition

Normal Subgroups, Section Subsection

Operation Tables, Subsection

Operations, Section

Order

of elements of a finite cyclic group, Theorem

Order of a Group Element, Definition

Order of a Recurrence Relation, Definition

Ordering Diagram, Paragraph

Partial Ordering, Definition Definition

Partially ordered set, Definition

Partition, Definition

of a group by cosets, Theorem

Path Graph, Definition

Permutation, Definition Definition

Permutation Counting Formula, Theorem

Permutation Groups, Section

Permutations

Composition, Note

Phrase Structure Grammar, Definition

Pigeonhole Principle, Theorem

Planar Graph, Definition

Plane Graph, Definition

Polynomial

Irreducible, Definition

Polynomial Addition, Definition

Polynomial Code, Example

Polynomial Expression

Non-recursive)., Definition

Recursive definition, Definition

Polynomial Multiplication, Definition

Polynomial over a Ring, Definition

Polynomial Units, Theorem

Polynomials, Subsection

Polynomials and their evaluation, Subsection

Poset, Definition

Posets Revisited, Section

Power Series, Definition

Power Series Units, Theorem

Power Set, Definition

Power Set Cardinality Theorem, Theorem

Powers of Functions, Definition

Prim's Algorithm, Algorithm

Proper subset, Paragraph

Properties of Functions, Section

Properties of Operations, Subsection

Proposition, Definition

Range of a Function., Definition

Recognition Problem, Problem

Rectangular codes, Exercise

Recurrence Relation, Definition

Recurrence Relations

Solving, Subsection

Recurrence relations obtained from “solutions”, Subsection

Recursive Language, Definition

Recursive Searching, Subsection

Reducibile Polynomial, Definition

References, References

Reflexive Relation, Definition

Regular Grammar, Definition

Relation, Definition

Relation Notation, Paragraph

Relation on a Set, Definition

Relatively Prime, Definition

Right Distributive Property, Definition

Ring, Definition

Commutative, Definition

Ring Isomorphism., Definition

Ring with unity, Definition

Robinson, Andrew, Poem

Rooted Tree, Definition

Rooted Trees, Section

Row Equivalent Matrices, Definition

Rule Of Products, The, Subsection

SageMath Note

bridge hands, Subsection

Cartesian Products and Power Sets, Subsection

Functions, Subsection

Graphs, Subsection

Kruskal's Algorithm, Subsection

Matrix Diagonalization, Subsection

Matrix Exponential, Subsection

Matrix Reduction, Subsection

Modular Arithmetic, Subsection

Power Series, Subsection

Search in a Graph, Subsection

Sets, Subsection

Scalar Multiplication, Definition

Sequence, Definition

Sequences, Section

Operations on,, Definition

Recursively Defined, Subsection

Set of Functions Between Two Sets, Definition

Set-Builder Notation, Paragraph

Sheffer Stroke, Definition

Solution Set, Definition

Some General Properties of Groups, Section

Span, Definition

Spanning Subgraph, Definition

Spanning Tree, Definition

Spanning Trees, Section

Spindel, Howard, Poem

Strings over an Alphabet, Definition

Subgraph, Definition

Submonoid

Generated by a Set, Definition

Subsystem, Definition

Subsystems, Section

Summation Notation and Generalizations, Section

Surjection, Definition

Symmetric Difference, Definition

Symmetric Group, Definition

Symmetric Relation, Definition

Systems of Linear Equations, Section

Tautology, Definition

Three Utilities Puzzle, Paragraph

Tournament Graph, Definition

Transitive Closure, Definition

Transitive Relation, Definition

Transposition, Definition

Traveling Salesman Problem, The, Subsection

Traversals of Binary Trees, Subsection

Traversals of Graphs, Section

Tree, Definition

Truth Set, Definition

Unary Operation., Definition

Undirected Graph, Definition

Union, Definition

Units

of a ring, Definition

of Polynomial Rings, Theorem

of Power Series Rings, Theorem

Unity of a Ring, Definition

Universal Quantifier, Definition

Universe, Definition

Upper Bound, Definition

Weighted Graph, Definition

What Is a Tree?, Section

Zero Divisor, Definition