SUNY Geneseo Department of Mathematics
Monday, April 29
Math 239 01
Spring 2019
Prof. Doug Baldwin
Tuesday (tomorrow) 5:00 - 8:00 PM, Macvittie Ballroom.
Hosted by student group Activists Fighting Racial Oppression (AFRO).
RSVP by 10:00 AM Tuesday at https://docs.google.com/forms/d/e/1FAIpQLScATpC6NTbGh3LDd_gfYHzGTvZOGrMX3pN1kO1oh-h1Upu5qA/viewform?usp=sf_link
Monday, May 13, 12:00 Noon.
Comprehensive, but emphasizing material since 2nd hour exam (e.g., sets, functions, relations, infinite sets; problem sets 7 through 10).
Rules and format otherwise similar to hour exams, especially open-references rule.
2 1/2 hours but designed for about 2 hours, i.e., less time pressure
I’ll bring donuts and cider.
Would you like a roughly 1 hour review session on study day? Yes!
I’ll find sample questions
SOFIs have started!
Please fill them out. I do read them and apply the feedback where possible to future classes (like the mid-semester feedback).
Finish proving that the union of two finite sets is finite.
The idea is to construct a bijection from A ∪ B to ℕn for some n, but finding a function that’s really a bijection took several iterations. Here is the final formal proof, with its LaTeX source here.
Section 9.2.
Lots of unintuitive things happen when you start working with infinite sets. For example, the book shows that the set of naturals is equivalent to the set of odd naturals, even though the latter seems to only be half of the former.
Another famous example is Hilbert’s infinite hotel: suppose you own a hotel with a countably infinite number of rooms. While it’s nice that you never run out of rooms, it’s also frustrating that you always have lots of empty rooms. Then one day a countably infinite tour bus arrives, completely full of tourists wanting rooms. At last! The Countably Infinite Hotel is full! Just as you’re celebrating, another countably infinite bus, also completely full, pulls up. The shame of turning guests away because the Countably Infinite Hotel has no rooms left would be huge, but what can you do? Then inspiration strikes: you just need to move each current guest from their current room, say room i, to room 2i. Then all the odd-numbered rooms are available, and you can accommodate everyone on the new bus too.
The moral of this story is that infinite “numbers” (more precisely, the infinite cardinality ℵ0) don’t get any bigger when you add 2 of them together. And in fact, you can keep using the same trick to accommodate any number of countably infinite buses, so apparently multiplying infinite cardinalities by any finite number doesn’t make them bigger either.
Wednesday, we’ll look at some of these strange behaviors more formally.
Proving finiteness via bijections