Day 7, Mat385

The key contains multiple different approaches to each problem, which I'll want to point out. Today, as we talk about proof techniques, I want to emphasize that there's more than one way to do any proof.

A few selected comments:

• The problems all came right out of things we discussed on the highlight for section 1.2: just checking to see if anyone was paying attention (and most of you were!).

• Problem 1: several of you realized that we should really be talking about predicate wffs. We're doing these somewhat "informally" (like Lewis Carroll did, when he said "Babies are crazy" -- by which he meant to say "\((\forall x)(B(x) \longrightarrow C(x))\)"):

  In this case, one might have translated it this way: \[ (\forall x)\left(C(x) \rightarrow A(x)\right) \land A(i) \rightarrow C(i) \]

  Your first steps would be \[ \begin{array}{lc} {1. (\forall x)\left(C(x) \rightarrow A(x)\right)}&{hyp}\cr {2. A(i)}&{hyp}\cr {3. C(i) \rightarrow A(i)}&{1, ui}\cr \end{array} \] and now we're stuck.

• This is a practical example of the problem that Bill McKibben brought up last time. Some people might "reason" this way:

  If you blow up churches, you're a terrorist; you're a terrorist; therefore you blow up buildings.

  No, it may just be that you "...lie down in front of a police car and you're [declared] a terrorist."

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