
Abelian group, 178

Absolute value

in E^{1}, 59

in E^{n}, 136

in Euclidean space, 180

in a normed linear space, 183

Additive inverse
in E^{n}, 131

Additivity of the volume of intervals in E^{n}, 168

Angle

between two hyperplanes in E^{n}, 153

between two lines in E^{n}, 147

between two vectors in E^{n}, 142

Antisymmetry of set inclusion, 2

Archimedean field. See Field, Archimedean

Archimedean property, 85

Argument of complex numbers, 176

Arithmetic sequence, 43

Associative laws

of addition and multiplication, 52

of set union and intersection, 5

of composition of relations, 29

Axioms

of addition and multiplication, 52

of an ordered field, 52

of order, 53

completeness axiom, 80

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Basic unit vector in E^{n}, 130, 133

Bernoulli inequalities, 71

Binary operations, 26. See also Function

Binomial coefficient, 73

Pascal's law, 73

Binomial theorem, 73

Boundary of an interval in E^{n}, 166

Bounded set in an ordered field, 78

left, or lower, bound of a, 78

maximum and minimum of a, 79

right, or upper, bound of a, 78

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C (the complex numbers), 172

C^{n}, 179

dot product in, 179

Cancellation laws in a field, 56

Cantor's diagonal process, 47. See also Sets

Cartesian product of sets, 18, 70, 129. See also Relations

CauchySchwarz inequality

in E^{n}, 137

in Euclidean space, 180

Center of an interval in E^{n}, 166

Characteristic function, 27

Closed

interval in E^{1}, 79

interval in E^{n}, 165

line segment in E^{n}, 148

Closure

of addition and multiplication in a field, 52

of addition and multiplication of integers, 75

of arithmetic operations on rationals, 76

Codomain. See Range

Collinear

lines in E^{n}, 147

points in E^{n}, 147

vectors in E^{n}, 137

Commutative

group, 178

laws of addition and multiplication, 52

laws of set union and intersection, 5

Complement of sets. See Difference of sets

Completeness axiom, 80

Complete ordered field. See Field, complete ordered

Complete ordered set, 113

Completion

of an Archimedean field, 116

of an ordered set, 113

Complex field, 172 . See also Complex numbers.

Complex numbers, 172

argument of, 176

conjugate of, 173

geometric representation of, 175

imaginary numbers in, 173

imaginary part of, 172

modulus of, 176

de Moivre's formula, 177

multiplicative inverse of, 174

polar coordinates of, 175

real part of, 172

real points in, 173

trigonometric form of, 176

Composition of relations, 28

associativity of, 29

Conjugate of a complex number, 173

Contracting sequence of sets, 40

Convergent sequence of sets, 44

Convex sets in E^{n}, 150, 169

Coplanar

set of points in E^{n}, 154

vectors in E^{n}, 154

Correspondences. See Relations

Countable

set, 41, 44

union, 46

Cross product

determinant definition of, 150

of sets, 18, 70, 129. See also Relations

of vectors in E^{3}, 150

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Dedekind cut, 112

Dedekind's theorem, 121

Density of an ordered field, 61, 88

Determinant

definition of cross products, 150

definition of hyperplanes, 158

Diagonal of an interval in E^{n}, 165

Diagonal process, Cantor's, 47. See also Sets

Difference of field elements, 55

Difference of sets, 4

generalized distributive laws with respect to, 10

symmetric, 11

Directed line in E^{n}, 146

Direction angles of a vector in E^{n}, 143

Direction cosines

of a line in E^{n}, 146

of a vector in E^{n}, 143

Disjoint sets, 4

Distance

between a point and a hyperplane in E^{n}, 159

between a point and a line in E^{n}, 151

between two lines in E^{n}, 151

between two points in E^{n}, 139

in Euclidean space, 181

in a normed linear space, 185

Distributive laws

of addition and multiplication, 53

of set union and intersection, 5, 9

with set differences, 10

Division of field elements, 56

Division theorem, 74

quotient, 74

remainder, 74

Domain

of a relation, 16

of a function or mapping, 23

Dot product, 135, 179. See also E^{n}

Double sequence, 47

Duality laws, de Morgan's, 7. See also Sets

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E^{1} (the real numbers), 51

E^{n} (Euclidean nspace), 129

absolute value of a vector in, 136

additive inverse of a vector in, 131

angle between two vectors in, 142

basic unit vector in, 130, 133

CauchySchwarz inequality, 137

collinear vectors in, 137

convex sets in, 150, 169

coplanar set of points in, 154

coplanar vectors in, 154

difference of vectors in, 130

direction, 144

direction angles of a vector in, 143

direction cosines of a vector in, 143

distance between points in, 139

dot product of vectors in, 135

globe in, 150

hyperplane in, 152 (see also Hyperplane in E^{n})

inner product of vectors in, 135

intervals in, 165 (see also Intervals in E^{n})

length of a vector in, 136

line in, 145 (see also Line in E^{n})

line segment in, 147 (see also Line segment in E^{n})

linear combination of vectors in, 133

linear functionals on, 154

linearly dependent set of vectors in, 135

linearly independent set of vectors in, 135

magnitude of a vector in, 136

modulus of a vector in, 136

norm of a vector in, 136

normalized vector in, 144

origin in, 130

orthogonal vectors in, 142

perpendicular vectors in, 142

plane in, 152 (see also Hyperplane in E^{n})

position vector in, 130

product of a scalar and a vector in, 131

scalar multiple of a vector in, 131

scalars of, 130

sphere in, 150

sum of vectors in, 130

triangle inequality in, 137

unit vector in, 144

vectors in, 130

zerovector of, 130

Edgelengths of an interval in E^{n}, 165

Elements of sets, 1

Empty set, 1, 41

Endpoints

of an interval in E^{1}, 79

of an interval in E^{n}, 165

of a line segment in E^{n}, 148

Equality

of sets, 2

of relations, 28

Equivalence class, 33. See also Equivalence relation

Equivalence relation, 32

equivalence class, 33

consistency of an, 32

modulo under an, 32

partition by an, 34

quotient set by an, 33

reflexivity of an, 32

substitution property of an, 32

symmetry of an, 32

transitivity of an, 32

Euclidean nspace. See E^{n}

Euclidean space, 180

absolute value in, 180

CauchySchwarz inequality in, 180

distance in, 181

principle of nested intervals, 182

Existential quantifier, 12

Expanding sequence of sets, 40

Extended real numbers, 121

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Family of sets, 1, 6

Field, 54

associative laws of addition and multiplication, 52

binomial theorem, 73

cancellation laws, 56

closure laws of addition and multiplication, 52

commutative laws of addition and multiplication, 52

complex, 172

difference, 55

distributive law of addition over multiplication, 53

division, 56

existence of additive and multiplicative inverses, 52

existence of additive and multiplicative neutral elements, 52

factorials in a, 69

first induction law, 64

inductive sets in a, 63

integers in a, 74

Lagrange identity, 141

natural elements in a, 63

powers in a, 69

quotient, 55

rationals in a, 75

subtraction, 56

Field, Archimedean. 85. See also Field, ordered

density of rationals in an, 88

integral part of an element of an, 87

Field, complete ordered. See also Field, Archimedean

Archimedean property of a, 85

completeness axiom, 80

definition of a, 81

greatest lower bound (g.l.b.), 80

infimum (inf), 80

isomorphism of, 104

least upper bound (l.u.b.), 80

powers in a, 94

roots, 90

supremum (sup), 80

Field, ordered, 54. See also Field

Archimedean field, 85

absolute value, 59

Bernoulli inequalities, 71

bounded sets in an, 78 (see also Bounded sets)

density of an, 61

division theorem, 74

inductive definitions in an, 39, 68

intervals in an, 78 (see also Interval)

irrational in an, 90

monotonicity, 53

negative elements of an, 54, 58

positive elements of an, 54, 58

prime numbers in an, 77

quotient of natural elements in an, 74

rational subfield of an, 76

rationals in lowest terms in an, 76

relatively prime integers in an, 76

remainder of natural elements in an, 74

second induction law, 67

transitivity, 53

trichotomy, 53

wellordering property of naturals in an, 67

Finite

sequence, 38

set, 41

Function, 23. See also Mapping

binary operations, 26

characteristic, 27

domain of a, 23

index notation or set, 25, 38

range of a, 23

value, 23

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Geometric representation of complex numbers, 175

Geometric sequence, 43

Globe in E^{n}, 150

Greatest lower bound (g.l.b.), 80

Group

Abelian, 178

commutative, 178

noncommutative, 178, 30

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Halfclosed

interval in E^{1}, 79

interval in E^{n}, 165

line segment in E^{n}, 148

Halfopen

interval in E^{1}, 79

interval in E^{n}, 165

line segment in E^{n}, 148

Hölder's inequality, 187. See also Normed linear space

Homomorphism, 105

Hyperplane in E^{n}, 152

angle between two hyperplanes, 153

coordinate equation of a, 152

determinant definition of a, 158

directed, 153

distance between a point and a, 159

linear functionals and, 154

normalized equations of a, 153

orthogonal projection of a point on a, 159

parallel hyperplanes, 153

pencil of hyperplanes, 159

perpendicular hyperplanes, 154

vector equation of a, 152

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Idempotent laws
of set union and intersection, 5

Identity map, 24

iff (if and only if), 3, 13

Image of a set under a relation, 17

Imaginary numbers in C, 173

Imaginary part of a complex number, 172

Inclusion relation of sets, 2

antisymmetry of, 2

reflexivity of, 2

transitivity of, 2

Index

notation, 6, 25, 38

sets, 6, 25

Induction, 63

first induction law, 64

induction law for integers in an ordered field, 75

inductive definitions, 39, 68

inductive hypothesis, 65

proof by, 64

second induction law, 67

Inductive

definitions, 39, 68

hypothesis, 65

proof, 64

set, 63

Infimum (inf), 80

Infinite sets, 41, 49, 45

Inner product, 135. See also E^{n}

Integers

closure of addition and multiplication, 75

in a field, 74

induction law for integers in an ordered field, 75

prime integers in an ordered field, 77

relatively prime integers in an ordered field, 76

Integral part, 87


Intersection

of sets, 4

of a family of sets, 6

Intervals in E^{1}, 78

closed, 79

endpoints of, 79

halfclosed, 79

halfopen, 79

open, 78

principle of nested, 85

Intervals in E^{n}, 165

additivity of volume of, 168

boundary of, 166

center of, 166

closed, 165

convexity of, 169

diagonal of, 165

edgelengths of, 165

endpoints of, 165

halfclosed, 165

halfopen, 165

open, 165

subadditivity of the volume of, 172

volume of, 166

Intervals of extended real numbers, 122

Inverse

image of a set under a relation, 17

function, map, or mapping, 24

relation, 16

Inverses,
existence of additive and multiplicative, 52

Invertible function, map, or mapping, 24

Irrational numbers, 47, 90, 119

Isomorphism, 104

isomorphic image, 104

of complete ordered fields, 104

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Lagrange identity, 141

Lagrange interpolation formula, 42

Least upper bound (l.u.b.), 80

Length

of an line segment in E^{n}, 148

of a vector in E^{n}, 136

Line in E^{n}, 145

angle between two lines, 147

directed, 146

direction cosines of a, 146

direction numbers of a, 146

distance between two lines in E^{n}, 151

nonparametric equations of a, 147

orthogonal projection of a point on a, 151

orthogonal projection of a vector on a, 149

parametric coordinate equations of a, 146

parametric equation of a, 146

Line segment in E^{n}, 147

closed, 148

endpoints of a, 148

halfclosed, 148

halfopen, 148

length of a, 148

open, 148

Linear

combination of vectors, 133, 179

equation, 152

functional, 154

mapping, 154, 179

space, 178 (see also Vector space)

Linearly dependent

set of vectors in E^{n}, 135

set of vectors in a vector space V, 179

Linearly independent

set of vectors in E^{n}, 135

set of vectors in a vector space V, 179

Logical quantifiers. See Quantifiers, logical

Lower limit

of a sequence of numbers, 123

of a sequence of sets, 44

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Magnitude of a vector in E^{n}, 136

Map. See Mapping

Mapping, 23. See also Function

as a relation, 23

identity, 24

inverse, 24

invertible, 24

linear, 154

onetoone, 23

onto, 23

Maximum of a bounded set, 79

Minkowski's inequality, 188. See also Normed linear space

Minimum of a bounded set, 79

Modulus

of a complex number, 176

of a vector in E^{n}, 136

de Moivre's formula, 177

Monotone

sequence of sets, 40

sequence of numbers, 40

strictly, 40

Monotonic, See Monotone

Monotonicity
of < with respect to addition and multiplication, 53

de Morgan's duality laws, 7

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Natural elements in a field, 63

Natural numbers, 55

and induction, 63

wellordering property of, 67

Negative numbers, 54, 58

Nested line segments, principle of

in E^{1}, 85

in Euclidean space, 182

in a normed linear space, 187

Neutral elements,
existence of additive and multiplicative, 52

Noncommutative group, 178, 30

Nonstandard analysis, 86

Norm

of a vector in E^{n}, 136

in a normed linear space, 183

Normalized vector in E^{n}, 144

Normed linear space, 183

absolute value in a, 183

distance in a, 185

Hölder's inequality, 187

Minkowski's inequality, 188

norm in a, 183

principle of nested line segments in a, 187

translation invariance of distance in a, 186

triangle inequality of distance in a, 186

triangle inequality of the norm in a, 183

Numbers

irrational, 47, 119

natural, 55

rational, 35, 46, 75, 119

real, 52 (see also Field, complete ordered

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Open

interval in E^{1}, 78

interval in E^{n}, 165

line segment in E^{n}, 148

Ordered

field, 54 (see also Field, ordered)

ntuple, 70, 3,
129

pair, 9, 3, 14, 38, 129

set, 53, 112

triple, 27, 129

Origin
in E^{n}, 130

Orthogonal projection

of a point on a line, 151

of a point on a hyperplane, 159

of a vector on a line, 149

Orthogonal vectors in E^{n}, 142

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Pair, ordered, 9, 3, 14, 38

inverse of, 15

Parallel

hyperplanes in E^{n}, 153

lines in E^{n}, 147, 150

vectors in E^{n}, 137, 150

Parametric coordinate equations of a line in E^{n},
146

Parametric equation of a line in E^{n}, 146

Pascal's law, 73

Pencil of hyperplanes, 159

Perpendicular

hyperplanes in E^{n}, 154

vectors in E^{n}, 142

Plane in E^{n}. See Hyperplane in E^{n}

Polar coordinates of complex numbers, 175

Position vector in E^{n}, 130

Positive numbers, 54, 58

Powers

with integer exponents, 69

with rational exponents, 94

with real exponents, 95

Prime

integers in an ordered field, 77

relatively, 76

Projection, orthogonal. See Orthogonal projection

Proof

by contradiction, 68

by induction, 64

Proper subset, 2

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Quantifiers, logical

existential, 12

negation of, 14

universal, 12, 14

Quotient

set by an equivalence relation, 33

of field elements, 55

of natural elements in an ordered field, 74

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Range

of a relation, 16

of a function or mapping, 23

Rationals

in a field, 75

in lowest terms in an ordered field, 76

Rational numbers, 119

countability of, 46

from natural numbers, 35

Rational subfield of an ordered field, 76

Real axis, 53

Real numbers. See also Field, complete ordered

binary approximations of, 100

construction of the, 111

decimal approximations of, 98

Dedekind cuts, 112

completeness axiom, 80

expansions of, 100

extended, 121

geometric representation of, 54

intervals of, 78

period of expansions of, 100

qary approximations of, 100

real axis, 53

terminating expansions of, 100

ternary approximations of, 100

Real part of a complex number, 172

Real points in C, 173

Reflexive relations, 17, 32

inclusion relation, 2

Relations, 14

as sets, 15

associativity of composition of, 29

composition of, 28

domain of, 16

equality of, 28

equivalence, 32 (see also Equivalence relations)

from Cartesian products of sets, 18

from cross products of sets, 18

image of a set under, 17

inverse of, 16

inverse image of a set under, 17

range of, 16

reflexive, 17, 32

symmetric, 17, 32

transitive, 17, 32

trichotomic, 17

Remainder (of natural elements in an ordered field), 74

Ring of sets, 172

Roots in a complete ordered field, 90, 91

Russell paradox, 11. See also Sets

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Scalar
of E^{n}, 130

Scalar multiple
in E^{n}, 131

Semiring of sets, 170

Seminorm, 184

Seminormed linear space, 184

Sequence, 38

arithmetic, 43

constant, 39

double, 47

finite, 38

geometric, 43

in index notation, 38

inductive definition of, 39

infinite, 38

lower limit of a, 123

as mappings, 38

monotone, 40

as ordered pairs, 38

strictly monotone, 40

subsequence, 40

upper limit of a, 123

Sets, 1

associative laws, 5

bounded sets in an ordered field, 78 (see also Bounded sets)

Cartesian products of, 18, 70

commutative laws, 5

complement of, 4

contracting sequence of, 40

convergent sequence of, 44

countable, 41, 44

countable union of, 46

cross products of, 18

difference of, 4

disjoint, 4

distributive laws, 5, 9, 10

duality laws, de Morgan's, 7

element of, 1

empty set, 1, 41

equality of, 2

expanding sequence of, 40

family of, 1, 6

finite, 41

idempotent laws, 5

index, 6

inductive, 63

infinite, 41, 49, 45

intersection of, 4

intersection of a family of, 6

lower limit of a sequence of, 44

monotone sequence of, 40

ordered, 53

proper subset of, 2

ring of, 172

Russell paradox, 11

semiring of, 170

subset of, 2

superset of, 2

symmetric difference of, 11

uncountable, 41, 45

union of, 4

union of a family of, 6

upper limit of a sequence of, 44

Venn diagrams, 5

Simple sets in E^{n}, 171

Sphere in E^{n}, 150

Strictly monotone sequences, 40

Subsequence, 40

Subadditivity of the volume of intervals in E^{n}, 172

Subset, 2

proper subset, 2

Subtraction of field elements, 56

Superset, 2

Supremum (sup), 80

Symmetric difference of sets, 11

Symmetric relations, 17, 32

Symmetries of plane figures, 31

as mappings, 31

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Transformation, 25. See also Mapping

Transitive relation, 17, 32

< as a, 53

inclusion relation, 2

Translation invariance of distance in a normed linear space, 186

Triangle inequality

in an ordered field, 60

in E^{n}, 137

of the distance in a normed linear space, 186

of the norm in a normed linear space, 183

Trichotomic relation, 17

< as a, 53

Trigonometric form of complex numbers, 176

Tuple (ordered), 70, 3

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Uncountable sets, 41, 45

Cantor's diagonal process, 47

irrational numbers, 47

Union

countable, 46

of sets, 4

of a family of sets, 6

Unit vector in E^{n}, 144

Universal quantifier, 12

Upper limit

of a sequence of numbers, 123

of a sequence of sets, 44

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Vector
in E^{n}, 130

Vector space, 178

complex, 179

normed linear space, 183 (see also Normed linear space)

real, 179

seminormed linear space, 184

Venn diagrams, 5. See also Sets

Volume of an interval in E^{n}, 166

additivity of the, 168

subadditivity of the, 172

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Wellordering property, 67

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Zerovector
in E^{n}, 130
