Professional Proofs

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• Questions?
  • Problem set is just about formally writing proofs, don’t need know-show tables
• Compare “Professional” Proofs to Our Guidelines
  • Examples: proof from “American Mathematical Monthly,” proof from Thomas’ Calculus
  • What are similarities?
    • Displayed equations
    • Titling and overall structure
    • End of proof marked by filled square or other special symbol
  • What are differences?
    • More flexible about format of displayed math
    • Skip restatement of proposition at beginning and end of proof
    • Detailed logic may not be explained in as much detail
      • (But the argument is still rigorously logical)
  • Why are there these differences?
    • Our guidelines emphasize novice proof-readers and -writers
• Next
  • Introduction to propositional logic, specifically common connectives
    • Connectives
      • And e.g., “7 > 3 and it is 9:10 AM”
        • Both propositions must be true
      • Or e.g., “7 > 3 or it is 9:10 AM”
        • At least 1 of the propositions must be true
        • Inclusive or vs exclusive or
      • “Nor” e.g., “neither is 7 > 3 nor is it 9:10 AM”?
        • Both propositions must be false
        • Build from “not”
          • e.g., “it is not 9:10 AM”
          • Proposition must be false
        • Nor could be “not ( ( 7 > 3 ) and (it is 9:10 AM) )”
          • Oops, this isn’t “nor” after all — it’s true if only one of its constituent propositions is false, it doesn’t require both of them to be false
        • Better “nor”: “( 7 is not greater than 3) and (it is not 9:10 AM)”
  • Read section 2.1

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