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Название: The Structure of the Real Number System
Авторы: Cohen L.W., Ehrlich G.
Аннотация:
The present course of mathematical education in school and college introduces a student rather casually to the various properties of integers, rational numbers, and real numbers as they are needed for arithmetic, algebra, geometry, and calculus. Sometimes the relations among these mathematical structures are sketched as indications of things to come in mathematics. When the student begins graduate study he finds that he is expected to be familiar with the structure of the real number HyHtem. If his graduate courses deal explicitly with this system at all, they usually do so in terms of a brief summary. It is then assumed that the student has a usable knowledge of the intricately interwoven properties of set theory, algebra and topology which characterize the Hystcm of real numbers. Actually, this assumption is frequently false and a gap is left in the student's knowledge which, if not filled, hinders his development.
Absolute value68 Accumulation point94 Addition of integers45 Addition of natural numbers21 Addition of rational numbers62 Addition of real numbers82 Anti-symmetric9 Archimedean order69 Associative15 Binary operation15 Binary relation9 Bounded sequence73 Cartesian product842 Categoricity58101 Choice, axiom of35 Closed set94 Co-domain11 Commutative15 Complete metric space90111 Complete ordered field85 Complex number106 Composition of mappings13 Congruence10 Convergence in metric spaces90 Convergence in ordered fields74 Couple41 Covering95 Cut71 Dense order69 Denumerable37 DIMENSION109 Disjoint sets6 Distributive15 Domain11 Element of a set2 Embedding56576684 Empty set4 Equivalence class10 Equivalence relation9436080106 Existence, Axiom of2 Factor set11 Field64 Finite set34 first element28 Function11 Fundamental sequence in a metric space89 Fundamental sequence in an ordered field73 Gap71 Generalized associative law30 Generalized commutative law32 Generalized Recursion Theorem19 Greatest lower bound93 Groupoid24 I-product42 I-tuple41 Identity element25 Identity, Axiom of2 Index set41 Induction, axiom of1617 Induction, Second Principle of29 Inductive set17 Infinite set37
Initial segment29 INTEGER44 Integral domain53 Intersection of sets5 interval93 Inverse element46 Inverse mapping13 Isometry91 Isomorphism56 Laurent series70101 Least upper bound93 LIMIT74 Lower bound92 Mapping11 Metric89110 Metric space88110 Multiplication of integers49 Multiplication of natural numbers23 Multiplication of rational numbers62 Multiplication of real numbers82 n-tuple41 Natural number16 Order for integers52 Order for natural numbers27 Order for rational numbers66 Order for real numbers84 Order relation11 Ordered field66 Ordered integral domain53 Ordered pair7 Ordered set11 pair5 Pairs, Axiom of5 Partial order relation11 Positive integer51 Positive rational number65 Positive real number83 Positive sequence77 Power set6 Powers, Axiom of6 Proper subset3 RANGE11 Rational number61 Real number81 Recursion theorem18 Reflexive9 Restriction of a binary operation15 Restriction of a mapping15 Ring50 Semigroup25 Set2 Singleton5 Specification, Axiom of3 Subset3 Successor16 symmetric9 Terminal segment34 Transitive9 Trichotomy9 Union of sets4 Unions, Axiom of4 Upper bound92 Vector space108