Appendix

Properties of Numbers

Throughout this book we have assumed an informal familiarity with the standardnumber systems used in high school mathematics. In this appendix we briefly sum-marize some of the commonly used properties of these number systems. A rigoroustreatment of these number systems, including proofs of everything stated in this ap-pendix, can be found in [Blo11, Chapters 1 and 2].

All the numbers we deal with in this book are real numbers. In particular, wedo not make use of complex numbers. We standardly think of the real numbers asforming the real number line, which extends infinitely in both positive and negativedirections. The real numbers have the operations addition, multiplication, negationand reciprocal, and the relations < and≤. (The real numbers also have the operationssubtraction and division, but we do not focus on them in this appendix because theycan be defined in terms of addition and multiplication, respectively.) Among the mostimportant properties of the real numbers are the following.

Theorem A.1. Let x, y and z be real numbers.

1. (x+ y)+ z = x+(y+ z) and (x · y) · z = x · (y · z) (Associative Laws).2. x+ y = y+ x and x · y = y · x (Commutative Laws).3. x+0 = x and x ·1 = x (Identity Laws).4. x+(−x) = 0 (Inverses Law).5. If x 6= 0, then x · x−1 = 1 (Inverses Law).6. If x+ z = y+ z, then x = y (Cancellation Law).7. If z 6= 0, then x · z = y · z if and only if x = y (Cancellation Law).8. x · (y+ z) = (x · y)+(x · z) (Distributive Law).9. −(−x) = x (Double negation).10. −(x+ y) = (−x)+(−y).11. (−x) · y =−(x · y) = x · (−y).12. If x < y and y < z, then x < z (Transitive Law).13. Precisely one of the following holds: either x < y, or x = y, or x > y (Tri-

chotomy Law).

© Springer Science+Business Media, LLC 2011

E.D. Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics, 341Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4419-7127-2,

342 Appendix: Properties of Numbers

14. If x≤ y and y≤ x, then x = y (Antisymmetry Law).15. x < y if and only if x+ z < y+ z.16. If z > 0, then x < y if and only if x · z < y · z.

We mention here two additional facts about the real numbers, which we will needin Section 7.8, though nowhere else. These facts involve the absolute value of realnumbers, which was defined in Exercise 2.4.9. A proof of the first of these facts maybe found in [Blo11, Lemma 2.3.9]; the second fact can be deduced from the firstwithout too much difficulty.

Theorem A.2. Let x,y ∈ R.

1. |x+ y| ≤ |x|+ |y| (Triangle Inequality).2. |x|− |y| ≤ |x+ y| and |x|− |y| ≤ |x− y|.

There are three particularly useful subsets of the real numbers, namely, the natu-ral numbers, the integers and the rational numbers.

The set of natural numbers is the set

N= {1,2,3,4, . . .}.

The sum and product of any two natural numbers is also a natural number, though thedifference and quotient of two natural numbers need not be a natural number. Beingreal numbers, the natural numbers satisfy all the properties of real numbers listedabove. The natural numbers also satisfy a number of special properties not satisfiedby the entire set of real numbers, for example the ability to do proof by induction;see Section 6.2 for more about the natural numbers.

We mention here one additional property of the natural numbers, which we willneed in Section 7.8, though again nowhere else. This property, rather than beingabout the natural numbers themselves, refers to the way that the natural numbers sitinside the real number.

Theorem A.3. Let x ∈ R. Then there is some n ∈ N such that x < n.

This theorem may seem intuitively obvious, but it is not trivial to prove, becauseits proof relies upon the Least Upper Bound Property of the real numbers. It wouldtake us too far afield to discuss the Least Upper Bound Property, but we will mentionthat it is the property of the set of real numbers that distinguish that set from theset of rational numbers; there is no difference between these two sets in terms ofalgebraic properties of addition, subtraction, multiplication and division. See [Blo11,Section 2.6] for a discussion of the Least Upper Bound Property in general, and aproof of Theorem A.3 in particular.

The set of integers is the set

Z= {. . .−3,−2,−1,0,1,2,3 . . .}.

The sum, difference and product of any two integers is also an integer, though thequotient of two integers need not be an integer. Being real numbers, the integerssatisfy all the properties of real numbers listed above.

Appendix: Properties of Numbers 343

We will need two additional properties of the integers; these properties do nothold for all real numbers. Our first property, given in the following theorem, is veryevident intuitively, though it requires a proof; see [Blo11, Exercise 2.4.4] for details.

Theorem A.4. Let a,b ∈ Z. If ab = 1, then a = 1 and b = 1, or a =−1 and b =−1.

Our second property of the integers, which is much less obviously true than theprevious property, is known as the Division Algorithm, though it is not an algorithm(the name is simply historical). See [Ros05, Section 1.5] for a proof.

Theorem A.5 (Division Algorithm). Let a,b ∈ Z. Suppose that b 6= 0. Then thereare unique q,r ∈ Z such that a = qb+ r and 0≤ r < |b|.

The set of rational numbers, denoted Q, is the set of all real numbers that canbe expressed as fractions. That is, a real number x is rational if x = a

b for someintegers a and b, where b 6= 0. Clearly, a rational number can be represented in morethan one way as a fraction, for example 1

2 = 36 . However, as we now state, there

is always a particularly convenient representation of each rational number, namely,writing it in “lowest terms.” This latter concept is phrased using the notion of integersbeing relatively prime, as defined in Exercise 2.4.3. The following theorem can beproved using the Fundamental Theorem of Arithmetic, which is found in [Ros05,Section 3.5]; a proof of the following theorem is also found in [Olm62, Section 402and Section 404].

Theorem A.6. Let x∈Q. Suppose that x 6= 0. There are a,b∈Z such that x = ab and

a and b are relatively prime. The integers a and b are unique up to negation.

It can be shown that the rational numbers are precisely those real numbers thathave decimal expansions that are either repeating, or are zero beyond some point;see [Blo11, Section 2.8] for a proof. The sum, difference, product and quotient ofany two rational numbers is also a rational number, except that we cannot divide byzero. The rational numbers are not all the real numbers; for example, the number

√2

is not rational, as is proved in Theorem 2.3.5. Again, being real numbers, the rationalnumbers satisfy all the properties of real numbers listed above.

The rational numbers also satisfy some additional nice properties, for example,they are “dense” in the real number line, which means that between any two realnumbers, no matter how close, we can always find a rational number; see [Blo11,Theorem 2.6.13] for a proof. We will rarely make use of such facts.

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Index

( f1, . . . , fn), 147−, 103ℵ0, 226⋂

X∈A , 112⋂i∈I , 112, 118⋃X∈A , 112⋃i∈I , 112, 1184, 108(n

k), 300

∩, 101∪, 101/0, 93≡ (mod n), 178{a, . . . ,b}, 200{a, . . .}, 200f : A→ B, 131GL2(R), 252, 264GL3(Z), 246glb, 275bxc, 177↔, 9f−1(Q), 141f (P), 140∨, 280∧, 5dxe, 152Z, 25¬, 7⇔, 19⇒, 17∨, 6lub, 275∧, 280

N, 936⊆, 95a, 70`, 70P (A), 98Pk(A), 302∏i∈I , 168Q, 93◦, 146R, 93∼, 222⊆, 95$, 97SL2(R), 264×, 104→, 8|A|, 232|x|, 69Zn, 180Z, 93Z-action, 169f n, 162, 214, 327f−1, 149f1×·· ·× fn, 147ut, xxi4, xxi///, xxi♦, xxi

Abel, Neils, 258absolute value, 69, 342absorption

law

352 Index

sets, 102absorption law, 282abstract algebra, 47, 50abuse of notation, 141AC, 121addition

logical implication, 17rule of inference, 27

adjunction, 27affirming the consequent, fallacy of, 31algebra

abstract, 47, 50boolean, 287

algebraic numbers, 241algebraic product, 331algebraic sum, 331algebraic topology, 258and, 5, 101, 205, 281antisymmetric relation, 271antisymmetry law

real numbers, 342argument

conclusion, 25consistent premises, 31inconsistent premises, 31logical, 25premises, 25valid, 26

Aristotle, 3, 61Arrow Impossibility Theorem, 270associative law, 253, 257

functions, 148lattices, 282logic, 19, 20real numbers, 341sets, 102

Axiom of Choice, 121, 122, 137, 159, 168,229, 237

Axiom of Choice for Disjoint Sets, 162axiomatic system, 47

backwards proof, 73, 84Bernoulli, Daniel, 217biconditional, 9biconditional-conditional, 17, 27bijective, 155, 222binary operation, 109, 251Binet’s formula, 217binomial coefficient, 244, 300

Binomial Theorem, 307, 308bivalence, 4boolean algebra, 287bound

greatest lower, 274, 280least upper, 127, 274, 280lower, 274upper, 127, 274

bound variable, 35bounded

sequence, 319Brouwer Fixed Point Theorem, 285Burnside’s Formula, 164

calculus, 74, 83, 129, 133, 146cancellation law

real numbers, 341canonical map, 182, 187, 265Cantor’s diagonal argument, 241Cantor, Georg, 92, 221, 226, 241, 244Cantor–Bernstein Theorem, 227cardinality, 232, 289

same, 222Cartesian product, 104cases, proof by, 64chain, 123

least upper bound, 127upper bound, 127

characteristic map, 139, 330China, 307choice function, 137

partial, 140closed bounded interval, 94closed unbounded interval, 94codomain, 131Cohen, Paul, 123, 244combinations, 300combinatorics, 288commutative diagram, 147, 198commutative law, 252, 257

lattices, 282logic, 19real numbers, 341sets, 102

complement, 331composite numbers, 61composition, 146, 262

associative law, 148identity law, 148

Index 353

noncommutativity, 148computer, 24, 204

programming language, 244science, 3, 212, 244, 276

conclusion, 25conditional, 8conditional-biconditional, 17, 27congruent modulo n, 178, 326conjunction, 5

associative law, 20commutative law, 19

consistent premises, 31constant, 133

map, 136sequence, 316

constantive, 328constructive dilemma, 17, 27Continuum Hypothesis, 244contradiction, 11

proof by, 58contrapositive, 20, 21

proof by, 58convergent

sequence, 314converges, 314converse, 21

fallacy of, 31coordinate function, 147countable set, 224countably infinite, 224counterexample, 76counting, 288covers, 272crystallography, 258

De Morgan’s law, 22logic, 20, 65sets, 104, 113

decimal expansion, 241definition by recursion, 198, 212denumerable set, 224denying the antecedent, fallacy of, 31derangement, 304derivation, 28determinant, 67, 252, 264diagram

commutative, 147, 198Venn, 101

dictionary order, 272

difference of set, 103direct proof, 54disjoint, 103

pairwise, 122disjunction, 6

associative law, 19commutative law, 19

distributive law, 105, 283logic, 20real numbers, 341sets, 102, 113

divergentsequence, 314

divides, 55, 324divisible, 55, 289, 294Division Algorithm, 163, 179, 186, 239, 290,

343domain, 131double negation, 19, 27, 63

real numbers, 341

element, 93greatest, 273, 287identity, 254inverse, 255least, 273, 287maximal, 123, 273minimal, 273

empty set, 93empty subset

fuzzy, 331epic, 155equality

functions, 136relations, 172sets, 97

equilateral triangle, 261equivalence, see logical equivalenceequivalence classes, 185equivalence relation, 185

and partitions, 188equivalent statements, 19Euclid, 47Euler, Leonhard, 71, 217even integers, 51excluded middle, law of the, xxii, 4, 63existence

and uniqueness, 74existential

354 Index

generalization, 41instantiation, 41quantifier, 36

extension function, 137

factor, 55factorial, 184, 214, 297fallacy, 31

of affirming the consequent, 31of denying the antecedent, 31of the converse, 31of the inverse, 31of unwarranted assumptions, 32

family of sets, 111indexed, 111

Fibonacci, 215numbers, 215, 311, 328sequence, 215, 313

finite set, 224first-order logic, 34fixed point, 145, 231, 285, 304fixed set, 164fraction, 55, 60, 93, 240, 343free variable, 35frieze pattern, 263function, 131

bijective, 155, 222composition, 146, 262constantive, 328coordinate, 147epic, 155equality, 136extension, 137fixed set, 164greatest integer, 144, 152, 177, 190hidempotent, 328image, 140injective, 155, 226, 235, 238inverse, 149

left, 148right, 148

inverse image, 141iteration, 162, 214, 327least integer, 152monic, 155nilpotent, 328one-to-one, 155onto, 155orbit, 163

order, 163order preserving, 277partial, 139, 229range, 140real-valued, 326relation preserving, 177, 339respects relation, 177restriction, 136set of, 164stabilizer, 163surjective, 155, 235, 238

Fundamental Theoremof Arithmetic, 209, 325, 343

fuzzyalgebraic product, 331algebraic sum, 331complement, 331empty subset, 331intersection, 331logic, 330set, 330subset, 330union, 331

Galileo, 221, 223, 225geometry, 258Godel, Kurt, 123, 244golden ratio, 217grammar, xx, xxiv, 82, 84greatest common divisor, 324greatest element

lattice, 287poset, 273

greatest integer function, 144, 152, 177, 190greatest lower bound

poset, 274, 280group, 257

abelian, 257homomorphism, 265isomorphism, 267subgroup, 260symmetry, 263trivial, 258

half-open interval, 94Haskell, 244Hasse diagrams, 272hidempotent, 328history of mathematics, xxi

Index 355

homomorphismgroup, 265join, 284meet, 284order, 276ring, 266

horses, 203hypothetical syllogism, 17, 27

idempotentlaw

sets, 102idempotent law, 282identity

element, 254law, 254, 257

functions, 148real numbers, 341sets, 102

map, 136matrix, 74

if and only if, 9theorems, 66

iff, see if and only ifimage, 140implication, see logical implicationimplies, 17inclusion map, 136inclusion-exclusion, principle of, 292inconsistent premises, 31indexed family of sets, 111induction, see mathematical inductioninductive

hypothesis, 203reasoning, 201step, 203

infinite, 224injective, 155, 226, 235, 238integers, 93

even, 51modulo n, 180odd, 51

intersection, 101, 112associative law, 102commutative law, 102fuzzy, 331

intervalclosed bounded, 94closed unbounded, 94

half-open, 94open bounded, 94open unbounded, 94

intuition, 47inverse

element, 255fallacy of, 31function, 149

left, 148right, 148

image, 141matrix, 75statement, 21

inverses law, 255, 257real numbers, 341

invertible matrix, 246, 252irrational numbers, 60isometry, 261isomorphic, 267isomorphism

group, 267order, 277

iteration, 162, 214, 327

join, 280homomorphism, 284

kernel, 267

lattice, 281complemented, 287distributive, 287greatest element, 287least element, 287

lawabsorption, 102, 282antisymmetry, 342associative, 19, 20, 102, 148, 253, 257,

282, 341cancellation, 341commutative, 19, 102, 252, 257, 282, 341De Morgan’s, 20, 22, 65, 104, 113distributive, 20, 102, 105, 113, 283, 341idempotent, 102, 282identity, 102, 148, 254, 257, 341inverses, 255, 257, 341of the excluded middle, xxii, 4, 63right inverses, 264transitive, 341

356 Index

trichotomy, 199, 341least element

lattice, 287poset, 273

least integer function, 152least upper bound

chain, 127poset, 274, 280

Least Upper Bound Property, 243, 275, 342left inverse function, 148Leonardo of Pisa, 215lexicographical order, 272limit, 221, 314linear ordering, 271, 276logic, 3

first-order, 34fuzzy, 330predicate, 34propositional, 34sentential, 34

logicalequivalence, 18implication, 15

logical argument, 25lower bound

greatest, 274, 280poset, 274

Lucasnumbers, 329sequence, 329

map, 131canonical, 182, 187, 265characteristic, 139, 330constant, 136identity, 136inclusion, 136projection, 137, 147

mathematicalnotation, xxiterminology, xxi

mathematical induction, 201principle of, 201

variant one, 206variant three, 208variant two, 208

mathematicshistory of, xxiphilosophy of, xxii

matrixdeterminant, 67, 252, 264identity, 74inverse, 75invertible, 246, 252trace, 67upper triangular, 68

maximal element, 123poset, 273

meet, 280homomorphism, 284

member, 93minimal element

poset, 273modus ponens, 17, 27modus tollendo ponens, 17, 27modus tollens, 17, 27monic, 155monotone, 145

nand, 25, 205natural numbers, 93, 197necessary, 9

and sufficient, 9negation, 205

logical, 7of statements, 21of statements with quantifiers, 39

nilpotent, 328not, 7null set, 93numbers

algebraic, 241composite, 61Fibonacci, 215, 311, 328irrational, 60Lucas, 329natural, 93, 197prime, 61, 71, 182, 184, 209rational, 60, 93real, 93

odd integers, 51one-to-one, 155onto, 155open bounded interval, 94open unbounded interval, 94operation

binary, 109, 251

Index 357

ternary, 257unary, 251

or, 6, 101, 205, 281orbit, 163order, 163

dictionary, 272homomorphism, 276isomorphism, 277lexicographical, 272partial, 271preserving function, 277total, 271

ordered pair, 104ordering

linear, 271, 276partial, 271quasi, 279total, 271

pair, ordered, 104pairwise disjoint, 122partial choice function, 140partial function, 139, 229partial order, 271partial ordering, 271partially ordered set, 271, see posetpartition, 187

and equivalence relations, 188Pascal’s triangle, 307, 308Peano Postulates, 197, 201permutations, 298philosophy of mathematics, xxiiphyllotaxis, 215Pigeonhole Principle, 212pointwise addition, 327polynomial, 241poodle-o-matic, 25poset, 271

greatest element, 273greatest lower bound, 274, 280least element, 273least upper bound, 274, 280lower bound, 274maximal element, 273minimal element, 273upper bound, 274

power set, 98, 143, 165, 225cardinality, 98

predicate logic, 34

premises, 25consistent, 31inconsistent, 31

prerequisites, xxprime numbers, 61, 71, 182, 184, 209

infinitely many, 62principle

of inclusion-exclusion, 292of mathematical induction, 201

variant one, 206variant three, 208variant two, 208

product, 104, 168product rule, 289projection map, 137, 147proof, 47

backwards, 73, 84by cases, 64by contradiction, 58by contrapositive, 58direct, 54existence and uniqueness, 74two-column, xx, 1, 27, 49, 52

proper subset, 97propositional logic, 34puzzle, 220Pythagorean Theorem, xix, 50

quantifier, 34existential, 36in theorems, 70universal, 35

quantum mechanics, 258quotient set, 186

rabbits, 215range, 140rational numbers, 60, 93real numbers, 93real-valued function, 326recursion, 212recursive

description, 212reflection, 261reflexive relation, 173, 271relation, 172

antisymmetric, 271class, 173equality, 172

358 Index

equivalence, 185on, 172reflexive, 173, 271symmetric, 173transitive, 173, 271

relation preserving function, 177, 339relatively prime, 69, 257, 325, 343repetition, 27respects relation, 177restriction function, 136right inverse function, 148right inverses law, 264ring, 266

homomorphism, 266rotation, 261rule

product, 289sum, 291

rules of inference, 26Russell’s Paradox, 115, 120Russell, Bertrand, 122

Schroeder–Bernstein Theorem, 227, 231,247

sentential logic, 34sequence, 165, 313

bounded, 319constant, 316Fibonacci, 215, 313Lucas, 329

set, 93countable, 224countably infinite, 224denumerable, 224difference, 103disjoint, 103element, 93empty, 93equality, 97finite, 98, 224fuzzy, 330infinite, 98, 224intersection, 101member, 93null, 93of functions, 164partially ordered, 271power, 98, 143, 165, 225product, 104, 168

quotient, 186subset, 95

proper, 97symmetric difference, 108, 264totally ordered, 271uncountable, 224union, 101

set difference, 103sets

absorption law, 102associative law, 102commutative law, 102De Morgan’s law, 104, 113distributive law, 102, 113family of, 111idempotent law, 102identity law, 102intersection, 112union, 112

simplification, 17, 27square root, 60stabilizer, 163statement, 4

equivalent, 19meta, 15

subgroup, 260trivial, 261

subset, 95empty

fuzzy, 331fuzzy, 330proper, 97

sufficient, 9sum rule, 291surjective, 155, 235, 238switching circuits, 24, 204symbols, xxisymmetric

difference, 108, 264relation, 173

symmetry, 261group, 263

tautology, 11term, 313ternary operation, 257TFAE, see the following are equivalentThales of Miletus, 47the following are equivalent, 67

Index 359

TheoremArrow Impossibility, 270Binomial, 307, 308Brouwer Fixed Point, 285Cantor–Bernstein, 227Pythagorean, xix, 50Schroeder–Bernstein, 227, 231, 247Wilson’s, 184

theoremif and only if, 66

topological sorting, 276total order, 271total ordering, 271totally ordered set, 271trace, 67transition course, xivtransitive law

real numbers, 341transitive relation, 173, 271Triangle Inequality, 342trichotomy law, 199, 341Trichotomy Law for Sets, 126, 229trivial subgroup, 261truth table, 5two-column proofs, xx, 1, 27, 49, 52

unary operation, 251uncountable set, 224union, 101, 112

associative law, 102commutative law, 102

fuzzy, 331uniqueness, 74universal

generalization, 41instantiation, 41quantifier, 35

unwarranted assumptions, fallacy of, 32upper bound

chain, 127least, 127, 274, 280poset, 274

upper triangular matrix, 68

valid argument, 26variable, 56, 85, 133

bound, 35free, 35

Venn diagram, 101

wallpaper pattern, 263well-defined, 135Well-Ordering Principle, 200Well-Ordering Theorem, 126, 280Wilson’s Theorem, 184writing mathematics, xx, xxiv, 80

Zeno, 195Zermelo–Fraenkel Axioms, 92, 111, 116,

197, 244ZF, 116ZFC, 121Zorn’s Lemma, 124, 229

44 4Ethan

Bloch wasborn in 1956,

and spent part ofhis childhood in Con-

necticut and part in Is-rael. He received a B.A. in

mathematics in 1978 from ReedCollege, where he developed a firm

belief in the value of a liberal arts edu-cation, and a Ph.D. in mathematics in 1983

from Cornell University, under the supervisionof Prof. David Henderson. He was an Instructor

at the University of Utah for three years, and arrivedat Bard College in 1986, where he has, very fortunately,

been ever since. He is married and has two children; hisfamily, his work and travel to Israel more than fill his time.

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