Why Read This Book? - Index of

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(a) {(x, y) : x ≤ y} (b) {(x, y) : x2 + y ≤ 4} (c) {(x, y) : |x| < 1} (d) {(x, y) : x2 + y2 ≥ 1} 3.11 Relations in General 113 You might think there is nothing particularly exciting about a relation, since any collection of ordered element pairs qualifies as a relation. What makes the idea take shape and become mathematically important is that we can lay out certain criteria by which pairs are included in the relation, and these different criteria might have particular properties that make interesting statements about A. Example 3.11.5 Define a relation R ⊆ Q × Q by R ={(p/q, r/s) : ps = qr}. This relation consists of all pairs of rational numbers that are equivalent, as was defined in Definition 3.8.2. So (3/8, −30/−80) and (0/2, 0/12) are in R, while (2/5, 5/3) is not. � Example 3.11.6 R1 ={(a, b) : a
114 Chapter 3 Sets and Their Properties Definition 3.11.10 Suppose R ⊆ A × A has the following properties. (E1) For all x ∈ A, (x, x) ∈ R. (Reflexive) (E2) For all x, y ∈ A,if(x, y) ∈ R, then (y, x) ∈ R. (Symmetric) (E3) For all x, y, z ∈ A,if(x, y), (y, z) ∈ R, then (x, z) ∈ R. (Transitive) Then R is called an equivalence relation on A. Notice how E1–E3 here say the same things as E1–E3 from Definition 3.6.1, but in different language. Instead of saying that x ≡ x for all x ∈ A is a true statement, we say that all ordered pairs of the form (x, x) are in the relation. Instead of saying that the truth of x ≡ y implies the truth of y ≡ x, now we say that the inclusion of the ordered pair (x, y) in the relation is always accompanied by the inclusion of (y, x). And instead of saying x ≡ y and y ≡ z implies x ≡ z, we say that the presence of (x, y) and (y, z) in the relation is always accompanied by the presence of (x, z). Thus an equivalence relation on A is a subset of A × A with some special properties that motivate a partition of A. Anytime we define a term that describes how elements of a set may or may not compare to each other, we can use that basis of comparison as a criterion for inclusion in a relation. For example, the symbol ≤ denotes a way that two real numbers relate or compare to each other. In Exercise 3.11.4(a), the statement x ≤ y is a criterion for the inclusion of a point (x, y) in a relation. The relation defined by ≤ has some special properties that motivate a definition of another type of relation. Since it is common to write xRy as a quicker way to write (x, y) ∈ R, and to say that x is related to y instead of saying (x, y) is an element of the relation, we use this somewhat more efficient notation in the next definition and example. Definition 3.11.11 properties. Suppose R is a relation on a set A with the following (O1) For all x ∈ A, xRx. (Reflexive) (O2) For all x, y ∈ A,ifxRy and yRx, then x = y. (Antisymmetric) (O3) For all x, y, z ∈ A,ifxRy and yRz, then xRz. (Transitive) Then R is called an order relation on A,orapartial ordering of A. Example 3.11.12 Show that the relation in Exercise 3.11.4(a) is an order relation on the real numbers. Solution We show that R has properties O1–O3. (O1) Since x = x for all x ∈ R, we have that x ≤ x, so that xRx for all x. (O2) Suppose xRy and yRx. Then x ≤ y and y ≤ x, so that x = y.
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A Transition to Abstract Mathematic
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A Transition to Abstract Mathematic
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For Topo my little mouse
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Contents Why Read This Book? xiii P
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3.10 Irrational Numbers 107 3.11 Re
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8.2 Subgroups 252 8.2.1 Subgroups D
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Why Read This Book? One of Euclid
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Preface A Transition to Abstract Ma
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Preface to the First Edition This t
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Preface to the First Edition xix ea
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Acknowledgments It takes an entire
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Notation and Assumptions 0 Suppose
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0.2 Assumptions about the Real Numb
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0.2 Assumptions about the Real Numb
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0.2.3 Other Assumptions 0.2 Assumpt
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P A R T I Foundations of Logic and
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Language and Mathematics 1 One main
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1.1 Introduction to Logic 13 Now im
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The compound statement we call “p
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1.1 Introduction to Logic 17 exampl
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1.2 If-Then Statements 19 Definitio
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1.2 If-Then Statements 21 (h) The o
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1.2 If-Then Statements 23 (g) If ei
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p : Meghan is at least 25 years old
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1.3 Universal and Existential Quant
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1.4 Negations of Statements 33 2. F
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1.4 Negations of Statements 35 want
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Solution 1.4 Negations of Statement
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Solution 1. Everyone in this class
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1.5 How We Write Proofs 41 Our purp
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1.5 How We Write Proofs 43 the nega
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Properties of Real Numbers 2 It’s
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2.1 Basic Algebraic Properties of R
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2.1.2 Properties of Multiplication
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2.2 Ordering Properties of the Real
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(c) For all real numbers a, a2 ≥
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2.3 Absolute Value 55 (⇐) Now sup
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2.4 The Division Algorithm 57 mn =
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2.5 Divisibility and Prime Numbers
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2.5 Divisibility and Prime Numbers
Page 86 and 87: Sets and Their Properties 3 All of
Page 88 and 89: U A B Figure 3.5 Venn diagram illus
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Page 92 and 93: 3.2 Proving Basic Set Properties 69
Page 94 and 95: 3.3 Families of Sets 71 Notice that
Page 96 and 97: Example 3.3.2 Consider the family o
Page 98 and 99: 3.3 Families of Sets 75 x ∈ � A
Page 100 and 101: 3.3 Families of Sets 77 and ... and
Page 102 and 103: Example 3.4.2 For any positive inte
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Page 106 and 107: Proof. To clean up the notation, we
Page 108 and 109: 3.5 Variations of the PMI 85 EXERCI
Page 110 and 111: (g) (a −m ) −n = a mn (h) (ab)
Page 112 and 113: Strong Induction 3.5 Variations of
Page 114 and 115: 3.6 Equivalence Relations 91 EXERCI
Page 116 and 117: 3.6 Equivalence Relations 93 beginn
Page 118 and 119: 3.6 Equivalence Relations 95 constr
Page 120 and 121: 3.7 Equivalence Classes and Partiti
Page 126 and 127: 3.8 Building the Rational Numbers 1
Page 128 and 129: 3.8 Building the Rational Numbers 1
Page 130 and 131: 3.10 Irrational Numbers 107 EXERCIS
Page 132 and 133: 3.10 Irrational Numbers 109 To the
Page 134 and 135: EXERCISE 3.10.2 √ 2 is irrational
Page 138 and 139: 3.11 Relations in General 115 (O3)
Page 140 and 141: 3.11 Relations in General 117 at al
Page 142 and 143: Functions 4 Second only to sets, fu
Page 144 and 145: 4.1 Definition and Examples 121 pro
Page 146 and 147: Solution We show that T satisfies p
Page 148 and 149: 4.2 One-to-one and Onto Functions 4
Page 150 and 151: 4.2 One-to-one and Onto Functions 1
Page 152 and 153: A 1 x A B f (A 1 ) Figure 4.4 The i
Page 154 and 155: 4.4 Composition and Inverse Functio
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Page 158 and 159: 4.5 Three Helpful Theorems 135 The
Page 160 and 161: 4.6 Finite Sets 137 EXERCISE 4.5.3
Page 162 and 163: 4.7 Infinite Sets 139 The next exer
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Page 166 and 167: 4.7 Infinite Sets 143 EXERCISE 4.7.
Page 168 and 169: EXERCISE 4.8.3 If A1,A2, and B are
Page 170 and 171: 4.8 Cartesian Products and Cardinal
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Page 174 and 175: 4.9 Combinations and Partitions 151
Page 180 and 181: 4.10 The Binomial Theorem 157 EXERC
Page 182 and 183: EXERCISE 4.10.3 Determine the follo
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P A R T I I Basic Principles of Ana
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The Real Numbers 5 Let’s look at
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5.1 The Least Upper Bound Axiom 167
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5.2 The Archimedean Property 169 EX
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5.2 The Archimedean Property 171 No
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5.3 Open and Closed Sets 173 Thus A
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5.4 Interior, Exterior, Boundary, a
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5.4 Interior, Exterior, Boundary, a
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5.5 Closure of Sets 179 The constru
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5.6 Compactness 181 EXERCISE 5.6.3
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5.6 Compactness 183 EXERCISE 5.6.13
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Sequences of Real Numbers 6.1 Seque
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6.1 Sequences Defined 187 Example 6
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EXERCISE 6.1.15 Consider the sequen
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L 1 e L L 2 e . . . . . . 1 2 3 4 N
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6.2 Convergence of Sequences 193 EX
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6.2 Convergence of Sequences 195 To
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6.3 The Nested Interval Property 19
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a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
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Example 6.4.7 The terms a0 = 1 a1 =
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Functions of a Real Variable 7 Func
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7.1 Bounded and Monotone Functions
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50 40 30 20 7 1 ε 7 7 2 ε 0 ( ( F
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7.2 Limits and Their Basic Properti
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7.2 Limits and Their Basic Properti
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7.3 More on Limits 217 values of 0
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7.4 Limits Involving Infinity 219 W
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7.4 Limits Involving Infinity 221 T
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(a) limx→a f(x) =−∞ (b) limx
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7.5 Continuity 225 If f is not cont
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1 2 3 4 5 6 Figure 7.6 Some example
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|a − x| |xa| = |x − a| |x||a| 7
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7.6 Implications of Continuity 231
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7.6 Implications of Continuity 233
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7.7 Uniform Continuity 235 we decla
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7.7 Uniform Continuity 237 Next, le
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7.7.2 Uniform Continuity and Compac
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P A R T I I I Basic Principles of A
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Groups 8 In its simplest terms, alg
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8.1 Introduction to Groups 245 are
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8.1 Introduction to Groups 247 Defi
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◦ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 0
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Show that C × is an abelian group
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8.2 Subgroups 253 G1 and G3 are aut
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8.2 Subgroups 255 Notice how proper
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8.2 Subgroups 257 Example 8.2.15 su
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EXERCISE 8.2.23 If G is a group and
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8.3 Quotient Groups 261 [0] ={...,
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8.3 Quotient Groups 263 is standard
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Step 2: Define an operation ∗H on
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(iii) (Z + 3.8) +Z (Z + 1.2) (iv)
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then for a given f ∈ S and any a
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Thus f 2 = f ◦ f = (132)(56) ◦
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2 3 1 3 1 Figure 8.1 Rigid square u
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EXERCISE 8.4.12 Complete Table 8.64
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8.5 Normal Subgroups 277 Theorem 8.
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8.5 Normal Subgroups 279 point to b
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8.6 Group Morphisms 281 EXERCISE 8.
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8.6 Group Morphisms 283 Notice that
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Ker � e x 2 x maps to x Ker � G
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Rings 9 We can create algebraic str
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9.1 Rings and Fields 289 (R9) Multi
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9.1 Rings and Fields 291 we define
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9.2 Subrings 293 In addition to pro
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9.2 Subrings 295 p1 = q1 = 3. Verif
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9.3 Ring Properties 297 Theorem 9.3
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9.3 Ring Properties 299 EXERCISE 9.
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9.4 Ring Extensions 301 in the inte
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9.4 Ring Extensions 303 Example 9.4
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9.4 Ring Extensions 305 We insist t
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9.5 Ideals 307 An ideal, say a left
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9.6 Generated Ideals 309 Proof. Sup
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EXERCISE 9.6.6 In the integers, wha
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9.7 Prime and Maximal Ideals 313 Ev
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9.8 Integral Domains 315 EXERCISE 9
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9.8 Integral Domains 317 Corollary
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9.9 Unique Factorization Domains 31
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9.10 Principal Ideal Domains 321 al
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9.10 Principal Ideal Domains 323 So
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9.11 Euclidean Domains 325 next exe
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9.11 Euclidean Domains 327 Exercise
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Proof. Let f and g be nonzero polyn
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9.12 Polynomials over a Field 331 W
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9.13 Polynomials over the Integers
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9.14 Ring Morphisms 335 Example 9.1
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9.14 Ring Morphisms 337 EXERCISE 9.
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9.15 Quotient Rings 339 this, howev
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9.15 Quotient Rings 341 Because the
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9.15 Quotient Rings 343 However, th
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Index =,51 ɛ (epsilon), 167-168 ɛ
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Index 347 Cosets, 263, 264, 265 Lag
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Index 349 Gauchy sequences, 202-206
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Index 351 identity, 124 relations v
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Index 353 Quaternions, 248, 253, 25
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Index 355 Tower of Hanoi, 84 Transc
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