We'll start class by going over the test. Grade distribution was A: 10, B: 4, C: 3, D: 2, and E: 11. I suspect the unfortunate preponderance of E's reflects those who did not read for understanding or work all the exercises :-(

A few shout outs and other things to discuss:

1. negating an implication
2. Brendan's elegant solution to problem 1
3. Luis's elegant solution to problem 2

To understand the negation of an implication better, run each of the following:

• ttg_cli.py "['P', 'Q']" -p "['P => Q', 'P => ~Q']" -i False
• ttg_cli.py "['P', 'Q']" -p "['P => Q', '~P => ~Q']" -i False
• ttg_cli.py "['P', 'Q']" -p "['P => Q', '~P or Q']" -i False
• ttg_cli.py "['P', 'Q']" -p "['P => Q', '~P or Q', '~(~P or Q)']" -i False
• ttg_cli.py "['P', 'Q']" -p "['~(P => Q)', 'P and ~Q']" -i False

Our friend José Ejemplo kindly shared his solutions to the test.

Before turning you loose on your homework, I promised to share one 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.

Complete the Practice Problems and Additional Exercises from Section 1.5: Proofs about Discrete Structures.

We will begin class with our first quiz on Section 1.3: Rules of Logic. After the quiz we will discuss Section 1.4: Proofs.

Complete the Practice Problems and as much of the Additional Exercises as time allows from Section 1.4.