I was out of town this day, so the class was covered by Joseph Vandehey. Here's his report on what he covered:

Today we covered implication and biconditionals.

For basic implication, we spent a long time going through the basics of why it should be true. We covered the basic truth table, then used examples like

P(x) : (2x = 8) => (x+2=6)

and noting that colloquially we would say this is true, and the statements P(4) and P(2) are true statements, despite the latter seeming very strange.

I also went through an example "Q: If it is raining outside, I will bring an umbrella to class" and talked about how to think of the implication as being a promise, and so long as the promise wasn't broken, then we would call the statement true. So even if it wasn't raining outside and I brought my umbrella, my promise held, so the statement was true. (This seemed to help the students a lot!)

In response to a student question, I quickly defined the contrapositive and went through its truth table.

We went through the definitions and truth tables for converses and biconditionals.

I also went through all the different ways we might see "P => Q" or "P <=> Q" written down, such as "P only if Q" and "P is a necessary and sufficient condition for Q"

I finished by going through some examples that showed that parentheses are needed with implications, namely because (P=>Q)=>R and P=>(Q=>R) had different truth tables.