Day 8, Mat385

Last time

Next time

Announcements
- Keep reading....
- We have a quiz at the end of the hour over predicate logic: section 1.3 and 1.4
Last time:
- We finished off Section 1.4, with a few comments from the quiz and problem #39.
- In particular we discovered that chatGPT hasn't quite learned predicate logic yet (couldn't do #39); in scraping the internet, it seems to have picked up some bad habits.
- Then it was on to Section 2.1: Proof techniques. We considered some alternative strategies, such as
  - Exhaustion (if there are only a finite number of cases)
  - Contraposition: \(P \rightarrow Q \leftrightarrow Q' \rightarrow P'\)
  - Contradiction: \(P \rightarrow Q \leftrightarrow (P \land Q' \rightarrow 0)\)
  - Serendipity!
Today:
- we'll hit section 2.2.
  (We'll only get through the "first" half today)
  - Induction is just dominoes.
  - While dominoes are usually finite in number (at least most of my sets are!), induction lets us prove a conjecture for an infinite number of things (usually positive integers -- i.e. natural numbers).
  - There are two types of induction proofs: first and second principle.
- Then we'll have a quiz, over sections 1.3 and 1.4: predicate logic.

Links:

Cayley's 1879 Note on the Four Color Problem:

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

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