- Page 3: Discrete Mathematicsfor New Technol
- Page 7 and 8: ContentsviiChapter 9: Boolean Algeb
- Page 9 and 10: xPreface to the Second EditionIn th
- Page 11 and 12: xiiPreface to the First Editionmath
- Page 13 and 14: xivPreface to the First Editionmanu
- Page 17 and 18: xviiiList of SymbolsO m×n the m ×
- Page 19 and 20: Chapter 1LogicLogic is used to esta
- Page 21 and 22: Logical Connectives and Truth Table
- Page 23 and 24: Logical Connectives and Truth Table
- Page 25 and 26: Logical Connectives and Truth Table
- Page 27 and 28: Logical Connectives and Truth Table
- Page 29 and 30: Logical Connectives and Truth Table
- Page 31 and 32: 1.3 Tautologies and ContradictionsT
- Page 33 and 34: Logical Equivalence and Logical Imp
- Page 35 and 36: Logical Equivalence and Logical Imp
- Page 37 and 38: Logical Equivalence and Logical Imp
- Page 39 and 40: The Algebra of Propositions 21Idemp
- Page 41 and 42: The Algebra of Propositions 23The d
- Page 43 and 44: Arguments 25As we have shown, ‘If
- Page 45 and 46: Arguments 27This shows that the arg
- Page 47 and 48: Predicate Logic 29A predicate descr
- Page 49 and 50: Predicate Logic 31The Existential Q
- Page 51 and 52: Predicate Logic 33the children didn
- Page 53 and 54: Predicate Logic 35Negation of Quant
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Predicate Logic 37(vii) If no-one w
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Arguments in Predicate Logic 392. U
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Arguments in Predicate Logic 41and
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Arguments in Predicate Logic 4310.
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Axioms and Axiom Systems 45to say,
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Axioms and Axiom Systems 47more the
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Methods of Proof 49There are two po
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Methods of Proof 51the truth of the
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Methods of Proof 53arbitrary even i
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Methods of Proof 55Examples 2.31. B
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Methods of Proof 57Examples 2.41. P
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Methods of Proof 59Suppose x is eve
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Methods of Proof 61constitute a pro
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Mathematical Induction 636. By prov
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Mathematical Induction 65Examples 2
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Mathematical Induction 67SolutionEm
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Mathematical Induction 69(This is p
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Mathematical Induction 71Now suppos
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Chapter 3Sets3.1 Sets and Membershi
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Sets and Membership 75D ={},theempt
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Sets and Membership 77Examples 3.31
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Subsets 79(vi) The set of integers
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Subsets 81When x = 1 2 ,2x 2 + 7x +
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Subsets 83If a universal set has be
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Operations on Sets 85(ii) Deduce th
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Operations on Sets 87Figure 3.2andA
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Operations on Sets 89Given a set A,
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Operations on Sets 91(b)In the foll
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Operations on Sets 934. Let Í ={1,
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Counting Techniques 95Counting Prin
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Counting Techniques 97SolutionLet
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The Algebra of Sets 99another. For
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The Algebra of Sets 101Figure 3.10T
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The Algebra of Sets 103(iii)(iv)(v)
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Families of Sets 105(ii)(iii)(iv)a
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Families of Sets 107collections of
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Families of Sets 109Power SetGiven
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Families of Sets 1113. Again we emp
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Families of Sets 113The first condi
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Families of Sets 1153. Which of the
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The Cartesian Product 117We are now
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The Cartesian Product 119Figure 3.1
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The Cartesian Product 1213. If X 1
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The Cartesian Product 123Theorem 3.
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The Cartesian Product 125Exercises
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The Cartesian Product 1277. (i) Def
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Types and Typed Set Theory 129Each
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Types and Typed Set Theory 131Boole
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Types and Typed Set Theory 133of pe
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Types and Typed Set Theory 135writi
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Types and Typed Set Theory 1373. n
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Types and Typed Set Theory 139Howev
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Types and Typed Set Theory 141is an
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Types and Typed Set Theory 143Exerc
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Types and Typed Set Theory 145(ix)
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Types and Typed Set Theory 147(iii)
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Relations and Their Representations
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Relations and Their Representations
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Relations and Their Representations
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Relations and Their Representations
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Relations and Their Representations
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Properties of Relations 159Examples
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Properties of Relations 161property
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Properties of Relations 163(ix)(x)
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Intersections and Unions of Relatio
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Intersections and Unions of Relatio
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Equivalence Relations and Partition
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Equivalence Relations and Partition
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Equivalence Relations and Partition
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Equivalence Relations and Partition
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Equivalence Relations and Partition
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Equivalence Relations and Partition
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Equivalence Relations and Partition
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Order Relations 183Definition 4.5A
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Order Relations 185Theorem 4.5Let R
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Order Relations 1872. Let A ={2, 3,
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Order Relations 1892. The alphabeti
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Order Relations 1917. Let be a non
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Hasse Diagrams 193Figure 4.6stateme
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Hasse Diagrams 195Figure 4.94. Show
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Hasse Diagrams 197ProofAs usual, we
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Application: Relational Databases 1
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Application: Relational Databases 2
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Application: Relational Databases 2
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Application: Relational Databases 2
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Application: Relational Databases 2
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Application: Relational Databases 2
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Application: Relational Databases 2
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Application: Relational Databases 2
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Definitions and Examples 215differe
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Definitions and Examples 217to B co
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Definitions and Examples 219second
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Definitions and Examples 221You are
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Definitions and Examples 223Figure
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Definitions and Examples 225Solutio
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Definitions and Examples 227Functio
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Definitions and Examples 229{4 if x
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Definitions and Examples 231we have
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Composite Functions 233(i)(ii)Figur
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Composite Functions 235Similarly,f
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Composite Functions 237Figure 5.11E
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Composite Functions 23910. Let f :
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Injections and Surjections 241Both
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Injections and Surjections 2432. Le
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Injections and Surjections 245Consi
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Injections and Surjections 247The e
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Injections and Surjections 249Proof
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Injections and Surjections 2514. De
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Injections and Surjections 2539. Le
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Bijections and Inverse Functions 25
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Bijections and Inverse Functions 25
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Bijections and Inverse Functions 25
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Bijections and Inverse Functions 26
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Bijections and Inverse Functions 26
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More on Cardinality 265Example 5.10
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More on Cardinality 267ProofThe pro
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More on Cardinality 269this type so
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Databases: Functional Dependence an
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Databases: Functional Dependence an
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Databases: Functional Dependence an
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Databases: Functional Dependence an
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Databases: Functional Dependence an
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Databases: Functional Dependence an
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Databases: Functional Dependence an
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Chapter 6Matrix Algebra6.1 Introduc
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Introduction 287The elements within
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Some Special Matrices 289termed the
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Operations on Matrices 291then mult
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Operations on Matrices 293Solution2
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Operations on Matrices 295Example 6
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Operations on Matrices 297⎛⎞=
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Operations on Matrices 299Solution(
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Operations on Matrices 3013. If(i)
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Elementary Matrices 303Example 6.7S
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Elementary Matrices 305What is inte
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Elementary Matrices 307The theorem
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Elementary Matrices 309will effect
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Elementary Matrices 311(i) find an
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The Inverse of a Matrix 313called s
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The Inverse of a Matrix 315ProofThe
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The Inverse of a Matrix 317Solution
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The Inverse of a Matrix 319In gener
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The Inverse of a Matrix 321Thus⎛1
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The Inverse of a Matrix 3233. If A,
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Chapter 7Systems of Linear Equation
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Introduction 327space, Ê 3 .A syst
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Introduction 329Example 7.1Write th
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Matrix Inverse Method 331Theorem 7.
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Matrix Inverse Method 333so that x
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Matrix Inverse Method 335system of
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Gauss-Jordan Elimination 337The fol
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Gauss-Jordan Elimination 339Example
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Gauss-Jordan Elimination 341Solutio
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Gauss-Jordan Elimination 343and the
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Gauss-Jordan Elimination 345Solutio
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Gauss-Jordan Elimination 347Solutio
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Gaussian Elimination 3492. 2x 1 + 7
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⎛⎜∼ ⎝⎛⎜∼ ⎝⎞1 1 1
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Gaussian Elimination 353a parameter
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Chapter 8Algebraic Structures8.1 Bi
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Binary Operations and their Propert
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Binary Operations and their Propert
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Binary Operations and their Propert
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Binary Operations and their Propert
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Algebraic Structures 365Definition
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Algebraic Structures 367Definition
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Algebraic Structures 369Testing fir
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Algebraic Structures 3712. If ∗ i
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More about Groups 37311. Let M deno
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More about Groups 375listed above a
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More about Groups 377Pre-multiplyin
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Some Families of Groups 379It is re
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Some Families of Groups 381Solution
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Some Families of Groups 383Symmetry
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Some Families of Groups 385A more c
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Some Families of Groups 387of compo
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Some Families of Groups 3898. Show
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Substructures 391From the table we
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Substructures 393The set A is close
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Substructures 395If (G, ∗) is a f
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Substructures 397(ii) (/8, + 8 )(ii
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Morphisms 399same even if the eleme
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Morphisms 401Alsof (x + y) = 2 x+y=
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Morphisms 403Isomorphism principleT
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Morphisms 405SolutionApplying theor
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Morphisms 407Theorem 8.10Let (A,
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Morphisms 409We havef (x + y) = 2(x
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Morphisms 4117. Let T ={A, B, C, D}
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Group Codes 413possible, then at le
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Group Codes 415word of length n con
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Group Codes 417Theorem 8.13A code i
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Group Codes 419Thus the sum of two
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Group Codes 421Example 8.18Consider
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Group Codes 423ProofIf w is a codew
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Group Codes 425distance is the mini
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Group Codes 427Definition 8.21If an
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Group Codes 4293. An encoding funct
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Chapter 9Boolean Algebra9.1 Introdu
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Introduction 433Definition 9.1 (con
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Properties of Boolean Algebras 435I
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Properties of Boolean Algebras 437P
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Properties of Boolean Algebras 439P
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Properties of Boolean Algebras 4412
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Boolean Functions 443Definitions 9.
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Boolean Functions 445a ‘rule’ f
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Boolean Functions 447ProofWe first
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Boolean Functions 449Theorem 9.10Ev
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Boolean Functions 451by substitutin
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Boolean Functions 453ProofThe metho
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Boolean Functions 455Example 9.5A B
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Boolean Functions 457From the examp
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(b) f i f j = E i E j for all f i ,
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Switching Circuits 461f (x 1 , x 2
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Switching Circuits 463switch we sha
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Switching Circuits 465equivalent to
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Switching Circuits 467an equivalent
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Logic Networks 469The AND-gate and
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Logic Networks 471diagram.In all th
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Logic Networks 473(i) (x 1 ⊕ x 2
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Minimization of Boolean Expressions
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Minimization of Boolean Expressions
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Minimization of Boolean Expressions
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Minimization of Boolean Expressions
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Minimization of Boolean Expressions
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Minimization of Boolean Expressions
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Chapter 10Graph Theory10.1 Definiti
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Definitions and Examples 489Figure
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Definitions and Examples 491Definit
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Definitions and Examples 493The com
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Definitions and Examples 495Note th
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Definitions and Examples 497(ii)(ii
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Definitions and Examples 499Describ
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Paths and Cycles 501Definitions 10.
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Paths and Cycles 503it as an easy e
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Paths and Cycles 505The following i
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Paths and Cycles 507The people of K
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Paths and Cycles 509Figure 10.10Alt
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Paths and Cycles 511(d)⎛⎜⎝1 1
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Paths and Cycles 51313. (i) Prove t
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Isomorphism of Graphs 515Figure 10.
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Isomorphism of Graphs 517ProofSee e
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Isomorphism of Graphs 519Isomorphis
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Trees 521give reasons to explain wh
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Trees 523Definition 10.12Let Ɣ be
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Trees 525(ii)Let e be any edge in T
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Trees 527(iii)How many non-isomorph
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Trees 52912. (i) How many spanning
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Planar Graphs 531Definition 10.13A
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Planar Graphs 533Examples 10.101. W
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Planar Graphs 535edges of both grap
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Planar Graphs 5376. Prove that, for
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Directed Graphs 539(‘ring doughnu
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Directed Graphs 541following, where
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Directed Graphs 543As we might expe
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Directed Graphs 545which has the gi
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Directed Graphs 547will need to lab
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Directed Graphs 549(iii)What is the
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Rooted Trees 551Before explaining o
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Rooted Trees 553Definitions 11.1A r
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Rooted Trees 555Definition 11.3Let
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Rooted Trees 557Definitions 11.4(i)
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Rooted Trees 559throughout the tree
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Rooted Trees 561(c) whether the tre
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Rooted Trees 5638. Let R be a parti
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Sorting 565expressions represented
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Sorting 567Example 11.4Suppose the
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Sorting 569perform each of the step
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Sorting 571right branch from Raven
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Sorting 573Step 2: List RavenStep 3
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Sorting 575(c). This has only moved
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Sorting 577Step 2: Obtaining the So
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Sorting 579The process of convertin
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Sorting 581to bottom, left to right
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Searching Strategies 583We shall co
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Searching Strategies 585Figure 11.1
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Figure 11.17Searching Strategies 58
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Searching Strategies 589Figure 11.1
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Weighted Graphs 5916. The following
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Weighted Graphs 593Figure 11.19Defi
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Weighted Graphs 595Algorithm 11.6 (
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Weighted Graphs 5972. An alternativ
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The Shortest Path and Travelling Sa
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The Shortest Path and Travelling Sa
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The Shortest Path and Travelling Sa
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The Shortest Path and Travelling Sa
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The Shortest Path and Travelling Sa
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The Shortest Path and Travelling Sa
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The Shortest Path and Travelling Sa
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Networks and Flows 613complete the
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Networks and Flows 615Figure 11.28o
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Networks and Flows 617The problem i
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Networks and Flows 619Figure 11.30D
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Networks and Flows 621Figure 11.32T
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Networks and Flows 623If we can fin
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Networks and Flows 625Activity Time
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Logic and Proof 627Logic and ProofF
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Boolean Algebra, Logic and Switchin
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Hints and Solutions to Selected Exe
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IndexAbel, Niels, 365Abelian group,
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Index 729inverse of an element with
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Index 731see Conjunctive normalform
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Index 733Dijkstra’s algorithm, 60
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Index 735leaf vertex of, 526Full ro
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Index 737Haken, Wolfgang, 551Half-a
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Index 739Leading element of a row o
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Index 741Monomorphism, 406Morphic i
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Index 743cardinality of, 112, 268,
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Index 745of a relation, 185Reverse
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Index 747Standard form of a linear
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Index 749of graphs, 499of relations