 Index of
Basic Concepts of Mathematics
by Elias Zakon

To indicate the range of topics covered in the electronic text Basic Concepts of Mathematics by Elias Zakon, we include here the book's index. According to the Terms and Conditions for the use of this text, it is offered free of charge to students using it for self-study and to teachers evaluating it as a required or recommended text for a course.

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
 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
 Copyright © 2001-2016 by The Trillia Group. The phrase "The Trillia Group" and The Trillia Group logo are trademarks of The Trillia Group. The Trillia Group participates in the eGranary Digital Library initiative of the WiderNet Project.