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Gentle Introduction to the Art of Mathematics

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Gentle Introduction to the Art of Mathematics

GIAM (a Gentle Introduction to the Art of Mathematics) is a free, open-source textbook that covers several topics in the foundations of mathematics (logic, sets, relations, functions and cardinality) and introduces the reader to many techniques of mathematical proof (direct, indirect, contradiction, contrapositive, mathematical induction, combinatorial proofs and magic)

Mathematical Logic and ProofMathematics
Author
Joseph Fields
Publisher
Southern Connecticut State University
Source
LibreTexts
License
GNU FDL
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 56 sections
  1. 01: Introduction and Notation
  2. 1.01: Basic Sets
  3. 1.02: Definitions - Prime Numbers
  4. 1.03: More Scary Notation
  5. 1.04: Definitions of Elementary Number Theory
  6. 1.05: Some Algorithms of Elementary Number Theory
  7. 1.06: Rational And Irrational Numbers
  8. 1.07: Relations
  9. 02: Logic and Quantifiers
  10. 2.01: Predicates and Logical Connectives
  11. 2.02: Implication
  12. 2.03: Logical Equivalences
  13. 2.04: Two-Column Proofs
  14. 2.05: Quantified Statements
  15. 2.06: Deductive Reasoning and Argument Forms
  16. 2.07: Validity of Arguments and Common Errors
  17. 03: Proof Techniques I
  18. 3.01: Direct Proofs of Universal Statements
  19. 3.02: More Direct Proofs
  20. 3.03: Indirect Proofs- Contradiction and Contraposition
  21. 3.04: Disproofs
  22. 3.05: Even More Direct Proofs- By Cases and By Exhaustion
  23. 3.06: Proofs and Disproofs of Existential Statements
  24. 04: Sets
  25. 4.01: Basic Notions of Set Theory
  26. 4.02: Containment
  27. 4.03: Set Operations
  28. 4.04: Venn Diagrams
  29. 4.05: 4.5 Russells Paradox
  30. 05: Proof Techniques II - Induction
  31. 5.01: The Principle of Mathematical Induction
  32. 5.02: Formulas for Sums and Products
  33. 5.03: Divisibility Statements and Other Proofs Using PMI
  34. 5.04: The Strong Form of Mathematical Induction
  35. 06: Relations and Functions
  36. 6.01: Relations
  37. 6.02: Properties of Relations
  38. 6.03: Equivalence Relations
  39. 6.04: Ordering Relations
  40. 6.05: Functions
  41. 6.06: Special Functions
  42. 07: Proof Techniques III - Combinatorics
  43. 7.01: Counting
  44. 7.02: Parity and Counting Arguments
  45. 7.03: The Pigeonhole Principle
  46. 7.04: The Algebra of Combinations
  47. 08: Cardinality
  48. 8.01: Equivalent Sets
  49. 8.02: Examples of Set Equivalence
  50. 8.03: Cantors Theorem
  51. 8.04: Dominance
  52. 8.05: The Continuum Hypothesis and The Generalized Continuum Hypothesis
  53. 09: Proof Techniques IV - Magic
  54. 9.01: Morleys Miracle
  55. 9.02: Five Steps Into the Void
  56. 9.03: Monges Circle Theorem