Day 8, Mat385

The Zooms continue

A menu of our past zooms...

• 19: sometimes the result seem so obvious that you wonder what to do! But can we say

  "2 consecutive integers are either odd and even or even and odd." Has that been proven? Is that a rule? It's easy enough to prove (a lot of these things are); but can we invoke it in a proof sequence?

  Many of you assumed that consecutive integers are one even, and one odd; I assumed that every integer is either even or odd (is that a theorem?). What theorems may we use, or rely on?

  In reality, we start with very few axioms, and some inference rules, and then begin developing theorems. Peano's axioms are the base of the number system.

  One of the hardest questions is "what's allowed?"

  Christian invoked a previous exercise (#15) -- good work!

  Here's Kyle's direct proof, with cases:

  

• 32: Nathan did a direct proof, using the prime factors of integers. Ryan and Kyle did nice contradiction proofs (slick! slicker than what I did). I (and many others) proved it by contraposition. If you do a contraposition proof, then you need to state the contraposition. Many of you turned an "and" into an "and", when it should have been an "or".

  Here's Ryan's proof:

  

• 71: Just choose a nice counterexample to show that it's false. You were handed a nice suggestion in #44.
• My work

On the Colouring of Maps
Author(s): Professor Cayley
Source: Proceedings of the Royal Geographical Society and Monthly Record of
Geography, New Monthly Series, Vol. 1, No. 4 (Apr., 1879), pp. 259-261