Finite and Infinite Sets

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• Misc
  • Review session
    • Tuesday, December 13, 3:00 - 4:30
    • Regular classroom
    • Bring questions
  • Final
    • Noon - 2:30, Wednesday December 14
    • Regular room
    • Comprehensive, but emphasizing material since 2nd hour exam (e.g., sets, functions, relations, infinite sets, etc.)
    • Designed for 1 1/2 - 2 hours, you’ll have 2 1/2
    • Style and rules otherwise similar to hour exams, especially open references
    • ca 6 - 8 questions, including some proofs
    • May also include question(s) applying proofs in contexts other than the ones seen in this course
    • Donuts and cider
  • SOFIs
    • 8 responses (42.11%) as of this morning
• Questions?
  • Best controller proofs?
    • Controller c: a ≤ c for all a in A
    • Best controller c: c ≤ x for all other controllers x
    • Proof by contradiction!
    • Assume better controller y ≠ c thus y < c, get contradiction
• Finite (and Not) Sets
  • Which is bigger — the set of natural numbers, the set of rational numbers, or the set of real numbers?
  • Section 9.1
  • Two sets A and B “are the same size” i.e., have same cardinality if and only if there is bijection between A and B
  • Show that ℕ ≈ ℕ ∪ {0}
    • Bijection: f: ℕ → ℕ ∪ {0} defined by f(n) = n - 1
  • Problem set
• Next
  • Countably infinite sets
  • Read section 9.2

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