Proofs and Concepts - The Fundamentals of Abstract Mathematics

by Dave Witte Morris & Joy Morris

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This textbook provides an introduction to proofs, logic, sets, functions, and other fundamental topics of abstract mathematics. It is designed to be the textbook for a bridge course that introduces undergraduates to abstract mathematics, but it is also suitable for independent study by undergraduates (or mathematically mature high-school students), or for use as a very inexpensive supplement to undergraduate courses in any field of abstract mathematics.

Description from LibreTexts.

Publisher
University of Lethbridge
Subjects
Mathematics, Mathematical Logic and Proof
Licence
CC BY-NC-SA 2.0, as LibreTexts states it. Non-commercial use only.
Who can read
Teens profiles and older, judged from its title and subjects

Contents

LibreTexts writes equations in TeX, and here they show as TeX; the PDF has them typeset.

  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Table of Contents
  5. Licensing
  6. 1: Propositional Logic
  7. 1.1: Assertions, Deductions, and Validity
  8. 1.2: Logic Puzzles
  9. 1.3: Using Letters to Symbolize Assertions
  10. 1.4: Connectives
  11. 1.5: Determining whether an assertion is true
  12. 1.6: Tautologies and contradictions
  13. 1.7: Logical Equivalence
  14. 1.8: Converse and Contrapositive
  15. 1.9: Some Valid Deductions
  16. 1.10: Summary
  17. 2: Two-Column Proofs
  18. 2.1: First Example of a Two-Column Proof
  19. 2.2: Hypothesis an Theorems in Two-column Proofs
  20. 2.3: Subproofs for \(\Rightarrow\)-introduction
  21. 2.4: Proof by Contradiction
  22. 2.5: Proof Strategies
  23. 2.6: What is a proof?
  24. 2.7: Counterexamples
  25. 2.8: Summary
  26. 3: Sets
  27. 3.1: Propositional Logic is Not Enough
  28. 3.2: Sets, Subsets, and Predicates
  29. 3.3: Operations on Sets
  30. 3.4: Summary
  31. PDF
  32. 4: First-Order Logic
  33. 4.1: Quantifiers
  34. 4.2: Translating to First-Order Logic
  35. 4.3: Negations
  36. 4.4: The Introduction and Elimination Rules for Quantifiers
  37. 4.5: Some Proofs about Sets
  38. 4.6: Counterexamples (reprise)
  39. 4.7: Summary
  40. 5: Sample Topics
  41. 5.1: Number Theory- Divisibility and Congruence
  42. 5.2: Abstract Algebra - Commutative Groups
  43. 5.3: Real Analysis - Convergent Sequences
  44. 5.4: Summary
  45. 6: Functions
  46. 6.1: Cartesian Product
  47. 6.2: Informal introduction to functions
  48. 6.3: Official Definition
  49. 6.4: One-to-one functions
  50. 6.5: Onto functions
  51. 6.6: Bijections
  52. 6.7: Inverse Functions
  53. 6.8: Composition of Functions
  54. 6.9: Image and Pre-Image
  55. 6.10: Summary
  56. 7: Equivalence Relations
  57. 7.1: Binary Relations
  58. 7.2: Definition and Basic Properties of Equivalence Relations
  59. 7.3: Equivalence Classes
  60. 7.4: Modular Arithmetic
  61. 7.5: Functions need to be Well-Defined
  62. 7.6: Partitions
  63. 7.7: Summary
  64. 8: Proof by Induction
  65. 8.1: The Principle of Mathematical Induction
  66. 8.2: Other Proofs by Induction
  67. 8.3: Other Versions of Induction
  68. 8.4: The Natural Numbers are Well-Ordered
  69. 8.5: Applications in Number Theory
  70. 8.6: Summary
  71. 9: Cardinality
  72. 9.1: Definition and Basic Properties
  73. 9.2: The Pigeonhole Principle
  74. 9.3: Cardinality of a Union
  75. 9.4: Hotel Infinity and the Cardinality of Infinite Sets
  76. 9.5: Countable sets
  77. 9.6: Uncountable sets
  78. 9.7: Summary
  79. Back Matter
  80. Index
  81. Glossary
  82. Detailed Licensing