Day 6, Mat385

Last time

Next time

Announcements
- I've added a folder with the edited highlights from class.
- Missed out on class last Thursday -- I hope that you all had a nice time sledding instead.
- I'm going to Zoom this session, but will stop by Thursday (unless there are health related reasons).
  Here's a menu of our past zooms...
- Saw this today, in the New York Times:
  The increasing inequality of the U.S. economy over the past half-century is unlikely to end because of a temporarily tight labor market. "Labor shortages may be a necessary condition for changing the nature of these [part-time] jobs," Noam says, "but they're generally not a sufficient condition." (What do you think: are labor shortages a necessary condition for changing the nature of these jobs?)
  
  $B$ is necessary for $A \iff A \rightarrow B$ $A$ is sufficient for $B \iff A \rightarrow B$ $B$ is necessary and sufficient for $A \iff A \leftrightarrow B$
  Logic is all around you!
- For assignment 1.3, which was due last night, I graded
  1. 8
    1. In any of these parts, if you don't specify a domain, it's wrong!
    2. You can't "split domains" ($x$ is a real, $y$ is an integer); you can add that to a predicate wff, however; in the domain of the reals, $P(y)$ is ("$y$ is an integer" $\land$ "$y$ is even").
    3. Part b was the trickiest. But look at all those implications! It's easy to concoct one where the antecedent is false, $(\forall x)A(x) \rightarrow (\forall x)B(x)$, by just making sure that it's not true that $(\forall x)A(x)$ in the domain; then choose $B(x)$ so that the consequent is true.
    4. Parts a and c were easier, because they were "$\land$"s, so you could just falsify one or the other of the parts of the wff....
  2. 9ab -- It's the "scope of the quantifier". Watch those parentheses, and brackets, however. I've noticed (e.g. on the next problem, #21), some falling down on the parenthesis matching.
  3. 21af
    - f. $(\forall x)\left(W(x) \land L(x) \rightarrow (\exists y)[A(x,y) \land J(y)] \right)$
      OR
    - f. $(\forall x)(\exists y)\left(W(x) \land L(x) \rightarrow A(x,y) \land J(y) \right)$
    This leads right into what we need to finish from section 1.4....
Last time:
- We finished up Section 1.3
- Then it was on to Section 1.4 (Predicate Logic), using our highlight worksheet 1.4
Today we'll finish off Section 1.4, and then hit highlight worksheet 2.1.

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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