Overview
We meet twice this week, and we will use that time to complete Chapter 1 Logic and Proofs. I will be out Thursday at a conference, so we need to plan for the full week in the beginning.
Thursday, February 26th
Classwork
Before turning you loose on your homework, I want to share a famous proof of
tremendous importance. Since we have so much to do, rather than give you a full
presentation, let me just share a two links.
Rational Numbers are Countably Infinite presents four different proofs that
the set of rational numbers and the set of natural numbers are the same
size
in that they can be put in a one-to-one correspondence.
The Wikipedia page
Cantor's
diagnonal argument is a proof that the set of real numbers is in some sense
bigger (infinitely bigger, actually), then the natural numbers and
integers are.
Classwork / Homework
Complete the Practice Problems and Additional Exercises from Section 1.5: Proofs about Discrete Structures.
Tuesday, February 24th
Classwork
We'll start class by going over the quiz. Grade distribution was A: 3, B+: 3, C: 5, D: 0, E: 0.
Which is a really good showing y'all, nice work!
Our friend José Ejemplo kindly shared his solutions to the quiz. Since he almost always gets a perfect score, we can use his solutions as an answer key.
A shout goes to Evan, who submitted something like this solution for the second proof:
| Statement | Justification |
|---|---|
| Assume P is true. | conditional proof assumption |
| Since P and P => Q are true, Q is true, and since P and P => R are true, R is true. | modus ponens |
| Since Q is true and R is true, Q and R is true. | logical conjunction |
| Therefore, P => (Q and R) | deduction theorem |
I plan to ask Evan whether he read ahead or has prior exposure to the study of logic, but it's nice that he is helping set us up for the next part of our study.
To better develop your understanding of the negation of an implication, run each of the following:
ttg_cli.py "['P', 'Q']" -p "['P => Q', 'P => ~Q']" -i Falsettg_cli.py "['P', 'Q']" -p "['P => Q', '~P => ~Q']" -i Falsettg_cli.py "['P', 'Q']" -p "['P => Q', '~P or Q']" -i Falsettg_cli.py "['P', 'Q']" -p "['P => Q', '~P or Q', '~(~P or Q)']" -i Falsettg_cli.py "['P', 'Q']" -p "['~(P => Q)', 'P and ~Q']" -i False
The rest of class time will be spent presenting as many of the exercises you chose last week as time permits.
Homework
Read Section 1.5: Proofs about Discrete Structure. You'll have both class and homework time on Thursday to complete the exercises.