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Properties of Real Numbers 2 It’s time to begin applying the language and logic of Chapter 1 to proof writing. In this chapter, you will learn to write proofs of important properties of real numbers that follow from assumptions A1–A22 outlined in Chapter 0. There are many principles of algebraic manipulation that you exploit without thinking. Here are some examples. If 2x + 3 = 13, then 2x = 10. If 2x = 10, then x = 5. If a
46 Chapter 2 Properties of Real Numbers The transitive property of equality allows us a convenience that we probably take for granted. When we write a = b = c = d, we interpret this as meaning that a = d. However, we must understand what is behind such a condensed statement. First, it includes the statement (a = b) ∧ (b = c), from which we conclude a = c by the transitive property. Second, it also includes the statement c = d, from which we conclude a = d, again by the transitive property. 2.1.1 Properties of Addition There are six properties of real number addition that we take as axioms. Here they are again for reference. (A2) Addition is well defined: For all a, b, c, d ∈ R, if a = b and c = d, then a + c = b + d. (A3) Closure property of addition: For all a, b ∈ R,a+ b ∈ R. The closure property also holds for N, W, Z, and Q. (A4) Associative property of addition: For all a, b, c ∈ R,(a+ b) + c = a + (b + c). (A5) Commutative property of addition: For all a, b ∈ R,a+ b = b + a. (A6) Existence of an additive identity: There exists a number 0 ∈ R with the property that a + 0 = a for all a ∈ R. (A7) Existence of additive inverses: For all a ∈ R, there exists some b ∈ R such that a + b = 0. Such an element b is called an additive inverse of a and is typically denoted −a. With these assumptions, we can now prove several properties of addition that follow from them. Read the proofs carefully, because you will mimic them to prove analogous properties for multiplication. Theorem 2.1.1 (Cancellation of addition). For all a, b, c ∈ R, if a + c = b + c, then a = b. Proof. Pick a, b, c ∈ R and suppose a + c = b + c. By A7, there exists −c ∈ R such that c + (−c) = 0. Since addition is well defined (A2), we have that [a + c]+(−c) =[b + c]+(−c), which by A4 we may write as a +[c + (−c)] =b + [c + (−c)]. This yields a + 0 = b + 0, which becomes a = b. We’re going to present a second proof of Theorem 2.1.1, written in a somewhat different style. The previous proof reveals the thought process involved in constructing the proof, but it uses several disjoint equations. The next proof shows how you might clean up these equations to create one extended equation that is the result we want.
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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
Page 18 and 19: Preface to the First Edition This t
Page 20 and 21: Preface to the First Edition xix ea
Page 22 and 23: Acknowledgments It takes an entire
Page 24 and 25: Notation and Assumptions 0 Suppose
Page 26 and 27: 0.2 Assumptions about the Real Numb
Page 28 and 29: 0.2 Assumptions about the Real Numb
Page 30 and 31: 0.2.3 Other Assumptions 0.2 Assumpt
Page 32 and 33: P A R T I Foundations of Logic and
Page 34 and 35: Language and Mathematics 1 One main
Page 36 and 37: 1.1 Introduction to Logic 13 Now im
Page 38 and 39: The compound statement we call “p
Page 40 and 41: 1.1 Introduction to Logic 17 exampl
Page 42 and 43: 1.2 If-Then Statements 19 Definitio
Page 44 and 45: 1.2 If-Then Statements 21 (h) The o
Page 46 and 47: 1.2 If-Then Statements 23 (g) If ei
Page 48 and 49: p : Meghan is at least 25 years old
Page 50 and 51: 1.3 Universal and Existential Quant
Page 56 and 57: 1.4 Negations of Statements 33 2. F
Page 58 and 59: 1.4 Negations of Statements 35 want
Page 60 and 61: Solution 1.4 Negations of Statement
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Page 64 and 65: 1.5 How We Write Proofs 41 Our purp
Page 66 and 67: 1.5 How We Write Proofs 43 the nega
Page 70 and 71: 2.1 Basic Algebraic Properties of R
Page 72 and 73: 2.1.2 Properties of Multiplication
Page 74 and 75: 2.2 Ordering Properties of the Real
Page 76 and 77: (c) For all real numbers a, a2 ≥
Page 78 and 79: 2.3 Absolute Value 55 (⇐) Now sup
Page 80 and 81: 2.4 The Division Algorithm 57 mn =
Page 82 and 83: 2.5 Divisibility and Prime Numbers
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Page 86 and 87: Sets and Their Properties 3 All of
Page 88 and 89: U A B Figure 3.5 Venn diagram illus
Page 90 and 91: (f) For all sets A, B, and C,ifA
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
Page 104 and 105: 3.4 The Principle of Mathematical I
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
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3.6 Equivalence Relations 95 constr
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3.7 Equivalence Classes and Partiti
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3.8 Building the Rational Numbers 1
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3.8 Building the Rational Numbers 1
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3.10 Irrational Numbers 107 EXERCIS
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3.10 Irrational Numbers 109 To the
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EXERCISE 3.10.2 √ 2 is irrational
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(a) {(x, y) : x ≤ y} (b) {(x, y)
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3.11 Relations in General 115 (O3)
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3.11 Relations in General 117 at al
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Functions 4 Second only to sets, fu
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4.1 Definition and Examples 121 pro
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Solution We show that T satisfies p
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4.2 One-to-one and Onto Functions 4
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4.2 One-to-one and Onto Functions 1
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A 1 x A B f (A 1 ) Figure 4.4 The i
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4.4 Composition and Inverse Functio
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4.4 Composition and Inverse Functio
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4.5 Three Helpful Theorems 135 The
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4.6 Finite Sets 137 EXERCISE 4.5.3
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4.7 Infinite Sets 139 The next exer
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4.7 Infinite Sets 141 of A. This or
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4.7 Infinite Sets 143 EXERCISE 4.7.
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EXERCISE 4.8.3 If A1,A2, and B are
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4.8 Cartesian Products and Cardinal
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4.8 Cartesian Products and Cardinal
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4.9 Combinations and Partitions 151
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4.10 The Binomial Theorem 157 EXERC
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EXERCISE 4.10.3 Determine the follo
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(c) The coefficient of ab 3 c 2 in
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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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