- 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
- 11 responses (57.89%) as of last night
- Section 9.2
- Countably infinite?
- 1-1 correspondence with naturals
- Set that’s infinite but members can be counted, count eventually reaches any member
- Countably infinite?
- Show that the set A = { x ∈ ℝ | sin x = 0 } is countably infinite
- sinx = 0 iff x = kπ for some integer k
- Correspondence: f:N→A defined by
- f(n) = nπ/2 if n is even
- f(n) = -(n-1)π/2 if n is odd
- Show that the set of prime numbers is countably infinite
- Use theorem that every subset of ℕ is countable?
- “Countable” = infinite but countable (countably infinite) or finite
- So you could use the theorem if you knew the primes was an infinite set
- Which it is, as follows:
- Suppose primes were finite, P = { p1, p2, … , pk }
- Recall that every n ≥ 2 in ℕ is prime or a product of primes
- P is infinite and subset of ℕ so therefore P is countably infinite
- Uncountable sets
- Read section 9.3