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Basic Language of Mathematics.
This book originates as an essential underlying component of a modern, imaginative three-semester honors program (six undergraduate courses) in Mathematical Studies. In its entirety, it covers Algebra, Geometry and Analysis in One Variable. The book is intended to provide a comprehensive and rigorou...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore :
World Scientific Publishing Company,
2014.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- PREFACE; Some symbols; CONTENTS; Index of terms; Index of names; Index of conditions; Index of symbols; Chapter 1. SETS; 11. Introduction; 12. Sets and their members; 13. Inclusion; 14. Set formation; 15. Special sets; 16. Basic operations; 17. Pairs; product sets; 18. Partitions; Chapter 2. MAPPINGS; 21. The concept of a mapping; 22. The graph of a mapping; 23. The range of a mapping; images and pre-images; the partition of a mapping; 24. Inclusion, identity, and partition mappings; 25. Composition of mappings; diagrams; restrictions and adjustments; 26. Mappings from a set to itself.
- Chapter 3. PROPERTIES OF MAPPINGS31. Constants; 32. Injective, surjective, and bijective mappings; 33. Inverses and invertibility; 34. Injectivity, surjectivity, and bijectivity: The induced mappings; 35. Cancellability; 36. Factorization; Chapter 4. FAMILIES; 41. The concept of a family; 42. Special families; 43. Families of sets; 44. Products and direct unions; 45. General associative and distributive laws; 46. Set-products and set-coproducts; Chapter 5. RELATIONS; 51. Relations in a set; 52. Images and pre-images; 53. Reversal, composition, and restriction of relations.
- 54. Relations from set to set functional relations; 55. Properties of relations; 56. Order; 57. Equivalence relations; Chapter 6. ORDERED SETS; 61. Basic concepts; 62. Isotone mappings; 63. Products; 64. Properties of ordered sets; 65. Lexicographic products and ordered direct unions; Chapter 7. COMPLETELY ORDERED SETS; 71. Completely ordered sets; 72. Pre-completely ordered sets; 73. Closure mappings; 74. Galois correspondences; 75. The fixed-point theorem for isotone mappings; Chapter 8. INDUCTION AND RECURSION; 81. Proof by induction; 82. Recursive definitions.
- Chapter 9. THE NATURAL NUMBERS91. Principles of counting; 92. Order; 93. General induction and recursive definitions; 94. Iteration; 95. Essential uniqueness of counting systems; 96. Addition and subtraction; 97. Multiplication and division; 98. Divisors and multiples; Chapter 10. FINITE SETS; 101. Finite sets and their cardinals; 102. Induction; 103. Operations with finite sets; 104. Factorials and binomial coefficients; 105. Orders in finite sets; 106. Finiteness without counting; Chapter 11. FINITE SUMS; 111. Commutative monoids; 112. Finite sums; 113. Sums of families with finite support.
- 114. Repeated and double sums115. Natural multiples; 116. The Inclusion-Exclusion Principle; 117. Sums in monoids of families; 118. Sums without zero; Chapter 12. COUNTABLE SETS; 121. Countable sets; 122. Some uncountable sets; 123. Another characterization of finiteness; Chapter 13. SOME ALGEBRAIC STRUCTURES; 131. Commutative monoids and groups; 132. Commutative rings; 133. Fields; Chapter 14. THE REAL NUMBERS: COMPLETE ORDERED FIELDS; 141. Introduction; 142. Ordered fields; 143. Complete ordered fields; 144. Essential uniqueness of complete ordered fields; Chapter 15. THE REAL-NUMBER SYSTEM.