Algebra

Автор(ы):Helena Peirse
06.10.2007
Год изд.:1970
Описание: This text is designed to give the student a background in the foundations of algebra and analysis. The algebra of symbolic logic and the concept of set are introduced early in the text so that the main definitional development of the complex number system flows easily from a set of postulates for the natural numbers. Important concepts are introduced when needed to provide better motivation through immediate usage. Thus, the approach is integrated for a greater continuity of ideas than would be possible if a course in sets were followed by a course in the number systems.
Оглавление:
Algebra — обложка книги. Обложка книги.
1 LOGIC
    Thales [1]
  1. Introduction [3]
  2. Statements and Connectives [5]
  3. Tautologies, Absurdities, and Contingencies [10]
  4. The Mathematics of Logic [14]
  5. Quantifiers [19]
  6. The Logic of Mathematics [24]
  7. A Simple Logical Discourse [29]
  8. Proof [34]
2 SETS
    Pythagoras [43]
  9. Sets and Subsets [45]
  10. Union and Intersection [51]
  11. Some Basic Properties [53]
  12. Complement [56]
  13. Power Set [59]
  14. Relations [62]
  15. Equivalence Relations [66]
  16. Functions [71]
  17. Operations [75]
3 BOOLEAN ALGEBRA
    Archimedes [83]
  18. Boolean Algebra [85]
  19. Logic [91]
  20. Switching Circuits [93]
4 NATURAL NUMBERS
    End of the Greek Era [103]
  21. Motivation [105]
  22. The Natural Numbers [110]
  23. Order [115]
  24. Subtraction and Division [118]
  25. Mathematical Induction [122]
  26. Exponents and Inequalities [128]
  27. Two More Principles [132]
  28. An Independent Postulate Set [134]
  29. The Peano Postulates [139]
  30. Cardinal Numbers and Addition [142]
  31. Cardinal Numbers and Multiplication [147]
5 INTEGERS
    Fibonacci [151]
  32. Ordered Pairs [153]
  33. Basic Properties [158]
  34. Inequality [163]
  35. Absolute Value [165]
  36. Mathematical Induction [169]
  37. The Integers [173]
6 RATIONAL NUMBERS
    Newton [177]
  38. Ordered Pairs [179]
  39. Identities [182]
  40. Subtraction and Division [184]
  41. Inequality [186]
  42. Miscellaneous Properties [190]
  43. The Field [194]
  44. Modular Arithmetic [198]
  45. Rings and Integral Domains [204]
7 REAL NUMBERS
    Gauss [211]
  46. Sequences and Limits [212]
  47. Cauchy Sequences [220]
  48. Equivalence, Addition, Multiplication [224]
  49. Basic Properties [227]
  50. Order [230]
  51. Completeness [233]
  52. Real Numbers [236]
  53. Dedekind Cuts [241]
8 COMPLEX NUMBERS
    Cauchy [247]
  54. Ordered Pairs [249]
  55. The Fundamental Theorem of Algebra [253]
9 INFINITY AND BEYOND
    Weierstrass [259]
  56. Infinity [261]
  57. Beyond Infinity [265]
  58. Cardinal Arithmetic [271]
  59. The March of the Ordinals [273]
  60. Three Paradoxes [276]
  61. Two Axioms [279]
APPENDICES]
  A. Umbugio [283]
  B. Proof of a Completeness Theorem [285]
  C. Synthetic Division [286]
  D. Proofs of Some Set Theorems [291]
HINTS FOR SELECTED EXERCISES [295]
ANSWERS FOR SELECTED EXERCISES [303]
BIBLIOGRAPHY [335]
INDEX [339]
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Язык:ENG
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