| Description |
1 online resource (321 pages) |
| Physical Medium |
polychrome |
| Description |
text file |
| Contents |
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. |
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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. |
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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. |
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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. |
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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. |
| Note |
151. The Real-Number System. |
| Summary |
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 rigorous account of the concepts on sets, mapping, family, order, number (both natural and real), as well as such distinct procedures like Proof by Induction and Recursive Definition, and the interaction between these ideas; with attempts at including insightful notes on historic and cultural settings and. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Mathematics.
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Mathematics. |
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Mathematical analysis.
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Mathematical analysis. |
| Genre/Form |
Electronic books.
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Electronic books.
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| Other Form: |
Print version: Schäffer, Juan Jorge. Basic Language of Mathematics. Singapore : World Scientific Publishing Company, ©2014 9789814596091 |
| ISBN |
9789814596107 (electronic book) |
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9814596108 (electronic book) |
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9789814596091 |
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