SUNY Geneseo Department of Mathematics
Wednesday, January 30
Math 239 01
Spring 2019
Prof. Doug Baldwin
There’s still room in the research weekend if anyone is still wondering whether to attend.
In the book’s exercises for section 1.1, why is “if x2 = 4 then x = 2” not a statement, while “for each real number x, if x2 = 4 then x = 2” is?
Because the first doesn’t say what x is, and for certain x’s (e.g., “my mother’s socks”) the rest wouldn’t even make enough sense to determine whether it’s true or false. But the second does say that x has to be a real number, which means that such things as squaring and equality do make sense, and so you can determine whether the rest is true or false. (It’s false, because there is a real number, namely -2, for which the hypothesis x2 = 4 is true, but the conclusion x = 2 is not.)
Section 1.2
(Progress check 1.9.1)
Prove that if x and y are even integers, then x + y is an even integer.
The longest part of this is writing it according to all the formal rules for proof-writing. Here’s the result:
Theorem: if x and y are even integers, then x + y is an even integer.
Proof. We assume that x and y are even integers, and will show that x + y is even. Thus x = 2n and y = 2p where n and p are integers. Then
x+y = 2n + 2p
= 2(n+p)
Since integers are closed under addition, n + p is an integer. Thus x + y = 2 times an integer, which means x + y is even by definition. QED
Note that the “forward questions” and “backward questions” Sundstrom talks about can be a very effective way to work out the logic of a proof, but they don’t appear explicitly in its written form.
Writing proofs formally looks like it’s going to be a pain. Introduce a computer-based tool called “LaTeX” that helps with some of the pain.
No reading, but if you want to try out LaTeX while I talk about it, bring a computer with access to LaTeX. LaTeX is free; see https://www.latex-project.org/get and the sites it links to for suggestions about versions for whatever computer you use (including places you can use LaTeX through a browser instead of putting it on your own computer at all).