Direct Proofs Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Certain proofs work very directly, by showing how a series of deductions leads from a hypothesis to a conclusion. The first part of section 1.2 in our textbook describes how you can often discover such proofs by asking and answering strategically chosen questions, using a table of deductions to organize your thinking.

Try using a similar process to prove some of the following:

1. If x and y are even integers, then xy is even.
2. If x and y are odd integers, then x + y is even.
3. If x is an even integer, and y is an odd integer, then x - y is odd.

I don’t expect any one person to prove all of these, and in fact I’d prefer if you didn’t. Offer some ideas, and then see what others build on those ideas. Eventually someone will put the last idea into a proof and finish it, but then people can turn their attention to the other proof(s).