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Proofs and Concepts - The Fundamentals of Abstract Mathematics

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Proofs and Concepts - The Fundamentals of Abstract Mathematics

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.

Mathematical Logic and ProofMathematics
Author
Dave Witte Morris & Joy Morris
Publisher
University of Lethbridge
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 73 sections
  1. 01: Propositional Logic
  2. 1.01: Assertions deductions and validity
  3. 1.02: Logic Puzzles
  4. 1.03: Using letters to symbolize assertions
  5. 1.04: Connectives
  6. 1.05: Determining whether an assertion is true
  7. 1.06: Tautologies and contradictions
  8. 1.07: Logical equivalence
  9. 1.08: Converse and contrapositive
  10. 1.09: Some valid deductions
  11. 1.10: Summary
  12. 02: Two-Column Proofs
  13. 2.01: First example of a two-column proof
  14. 2.02: Hypothesis an theorems in two-column proofs
  15. 2.03: Subproofs for introduction
  16. 2.04: Proof by contradiction
  17. 2.05: Proof strategies
  18. 2.06: What is a proof
  19. 2.07: Counterexamples
  20. 2.08: Summary
  21. 03: Sets
  22. 3.01: Propositional Logic is not enough
  23. 3.02: Sets subsets and predicates
  24. 3.03: Operations on sets
  25. 3.04: Summary
  26. PDF
  27. 04: First-Order Logic
  28. 4.01: Quantifiers
  29. 4.02: Translating to First-Order logic
  30. 4.03: Negations
  31. 4.04: The Introduction and elimination rules for quantifiers
  32. 4.05: Some proofs about sets
  33. 4.06: Counterexamples (reprise)
  34. 4.07: Summary
  35. 05: Sample Topics
  36. 5.01: Number theory- divisibility and congruence
  37. 5.02: Abstract Algebra- commutative groups
  38. 5.03: Real analysis- convergent sequences
  39. 5.04: Summary
  40. 06: Functions
  41. 6.01: Cartesian product
  42. 6.02: Informal introduction to functions
  43. 6.03: Official Definition
  44. 6.04: One-to-one functions
  45. 6.05: Onto functions
  46. 6.06: Bijections
  47. 6.07: Inverse functions
  48. 6.08: Composition of functions
  49. 6.09: Image and pre-image
  50. 6.10: Summary
  51. 07: Equivalence relations
  52. 7.01: Binary relations
  53. 7.02: Definition and basic properties of equivalence relations
  54. 7.03: Equivalence classes
  55. 7.04: Modular arithmetic
  56. 7.05: Functions need to be well-defined
  57. 7.06: Partitions
  58. 7.07: Summary
  59. 08: Proof by Induction
  60. 8.01: The Principle of mathematical induction
  61. 8.02: Other proofs by induction
  62. 8.03: Other versions of induction
  63. 8.04: The natural numbers are well-ordered
  64. 8.05: Applications in number theory
  65. 8.06: Summary
  66. 09: Cardinality
  67. 9.01: Definition and basic properties
  68. 9.02: The pigeonhole principle
  69. 9.03: Cardinality of a union
  70. 9.04: Hotel infinity and the cardinality of infinite sets
  71. 9.05: Countable sets
  72. 9.06: uncountable sets
  73. 9.07: Summary