-
Abelian group, 178
-
Absolute value
-
in E1, 59
-
in En, 136
-
in Euclidean space, 180
-
in a normed linear space, 183
-
Additive inverse
in En, 131
-
Additivity of the volume of intervals in En, 168
-
Angle
-
between two hyperplanes in En, 153
-
between two lines in En, 147
-
between two vectors in En, 142
-
Anti-symmetry 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
-
Back to Top
-
Basic unit vector in En, 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 En, 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
-
Back to Top
-
C (the complex numbers), 172
-
Cn, 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
-
Cauchy-Schwarz inequality
-
in En, 137
-
in Euclidean space, 180
-
Center of an interval in En, 166
-
Characteristic function, 27
-
Closed
-
interval in E1, 79
-
interval in En, 165
-
line segment in En, 148
-
Closure
-
of addition and multiplication in a field, 52
-
of addition and multiplication of integers, 75
-
of arithmetic operations on rationals, 76
-
Co-domain. See Range
-
Collinear
-
lines in En, 147
-
points in En, 147
-
vectors in En, 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 En, 150, 169
-
Coplanar
-
set of points in En, 154
-
vectors in En, 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 E3, 150
-
Back to Top
-
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 En, 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 En, 146
-
Direction angles of a vector in En, 143
-
Direction cosines
-
of a line in En, 146
-
of a vector in En, 143
-
Disjoint sets, 4
-
Distance
-
between a point and a hyperplane in En, 159
-
between a point and a line in En, 151
-
between two lines in En, 151
-
between two points in En, 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 En
-
Double sequence, 47
-
Duality laws, de Morgan's, 7. See also Sets
-
Back to Top
-
E1 (the real numbers), 51
-
En (Euclidean n-space), 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
-
Cauchy-Schwarz 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 En)
-
inner product of vectors in, 135
-
intervals in, 165 (see also Intervals in En)
-
length of a vector in, 136
-
line in, 145 (see also Line in En)
-
line segment in, 147 (see also Line segment in En)
-
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 En)
-
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
-
zero-vector of, 130
-
Edgelengths of an interval in En, 165
-
Elements of sets, 1
-
Empty set, 1, 41
-
Endpoints
-
of an interval in E1, 79
-
of an interval in En, 165
-
of a line segment in En, 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 n-space. See En
-
Euclidean space, 180
-
absolute value in, 180
-
Cauchy-Schwarz inequality in, 180
-
distance in, 181
-
principle of nested intervals, 182
-
Existential quantifier, 12
-
Expanding sequence of sets, 40
-
Extended real numbers, 121
-
Back to Top
-
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
-
well-ordering 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
-
Back to Top
-
Geometric representation of complex numbers, 175
-
Geometric sequence, 43
-
Globe in En, 150
-
Greatest lower bound (g.l.b.), 80
-
Group
-
Abelian, 178
-
commutative, 178
-
noncommutative, 178, 30
-
Back to Top
-
Half-closed
-
interval in E1, 79
-
interval in En, 165
-
line segment in En, 148
-
Half-open
-
interval in E1, 79
-
interval in En, 165
-
line segment in En, 148
-
Hölder's inequality, 187. See also Normed linear space
-
Homomorphism, 105
-
Hyperplane in En, 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
-
Back to Top
-
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
-
anti-symmetry 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 En
-
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 E1, 78
-
closed, 79
-
endpoints of, 79
-
half-closed, 79
-
half-open, 79
-
open, 78
-
principle of nested, 85
-
Intervals in En, 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
-
half-closed, 165
-
half-open, 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
-
Back to Top
-
Lagrange identity, 141
-
Lagrange interpolation formula, 42
-
Least upper bound (l.u.b.), 80
-
Length
-
of an line segment in En, 148
-
of a vector in En, 136
-
Line in En, 145
-
angle between two lines, 147
-
directed, 146
-
direction cosines of a, 146
-
direction numbers of a, 146
-
distance between two lines in En, 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 En, 147
-
closed, 148
-
endpoints of a, 148
-
half-closed, 148
-
half-open, 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 En, 135
-
set of vectors in a vector space V, 179
-
Linearly independent
-
set of vectors in En, 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
-
Back to Top
-
Magnitude of a vector in En, 136
-
Map. See Mapping
-
Mapping, 23. See also Function
-
as a relation, 23
-
identity, 24
-
inverse, 24
-
invertible, 24
-
linear, 154
-
one-to-one, 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 En, 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
-
Back to Top
-
Natural elements in a field, 63
-
Natural numbers, 55
-
and induction, 63
-
well-ordering property of, 67
-
Negative numbers, 54, 58
-
Nested line segments, principle of
-
in E1, 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 En, 136
-
in a normed linear space, 183
-
Normalized vector in En, 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
-
Back to Top
-
Open
-
interval in E1, 78
-
interval in En, 165
-
line segment in En, 148
-
Ordered
-
field, 54 (see also Field, ordered)
-
n-tuple, 70, 3,
129
-
pair, 9, 3, 14, 38, 129
-
set, 53, 112
-
triple, 27, 129
-
Origin
in En, 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 En, 142
-
Back to Top
-
Pair, ordered, 9, 3, 14, 38
-
inverse of, 15
-
Parallel
-
hyperplanes in En, 153
-
lines in En, 147, 150
-
vectors in En, 137, 150
-
Parametric coordinate equations of a line in En,
146
-
Parametric equation of a line in En, 146
-
Pascal's law, 73
-
Pencil of hyperplanes, 159
-
Perpendicular
-
hyperplanes in En, 154
-
vectors in En, 142
-
Plane in En. See Hyperplane in En
-
Polar coordinates of complex numbers, 175
-
Position vector in En, 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
-
Back to Top
-
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
-
Back to Top
-
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
-
q-ary 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
-
Back to Top
-
Scalar
of En, 130
-
Scalar multiple
in En, 131
-
Semi-ring of sets, 170
-
Semi-norm, 184
-
Semi-normed 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
-
semi-ring 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 En, 171
-
Sphere in En, 150
-
Strictly monotone sequences, 40
-
Subsequence, 40
-
Subadditivity of the volume of intervals in En, 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
-
Back to Top
-
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 En, 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
-
Back to Top
-
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 En, 144
-
Universal quantifier, 12
-
Upper limit
-
of a sequence of numbers, 123
-
of a sequence of sets, 44
-
Back to Top
-
Vector
in En, 130
-
Vector space, 178
-
complex, 179
-
normed linear space, 183 (see also Normed linear space)
-
real, 179
-
semi-normed linear space, 184
-
Venn diagrams, 5. See also Sets
-
Volume of an interval in En, 166
-
additivity of the, 168
-
subadditivity of the, 172
-
Back to Top
-
Well-ordering property, 67
-
Back to Top
-
Zero-vector
in En, 130
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