SUNY Geneseo Department of Mathematics
Monday, February 18
Math 239 01
Spring 2019
Prof. Doug Baldwin
The first hour exam is Friday, Feb. 22.
There is a sample exam (the first exam from last time I taught this course) available on Canvas.
Our exam will cover material from the beginning of the semester through problem set 3 (e.g., mathematical statements, direct proofs, propositional logic, sets, predicates).
Expect 3 to 5 short-answer questions, similar (in my mind at least) to problem set questions.
You’ll have the whole class period to do the test.
Open references (book, notes, online references), but closed person.
Section 3.1.
Conjecture: If n is an odd integer, then n ≡ 1 (mod 2).
Try out some examples to be sure you understand what this means in the first place, and to see if it seems plausible.
Definition of congruence to 1 modulo 2: 2 | (1-n) i.e., 2 divides into 1 - n.
Does this work for n = 1? 1 - n = 1 - 1 = 0. Does 2 | 0, i.e., is 0/2 an integer, i.e., is 0 = 2k for some integer k? Yes, k =0.
Does this work for n = 3? Is 1 - 3 divisible by 2, i.e., = 2k for integer k? 1 - 3 = -2, so k = -1.
How about n = 6, i.e., does 2 divide 1 - 6, i.e., does 2 divide -5? No because while -5 = 2 × (-2.5), there’s no integer k such that -5 = 2k.
Can you prove it?
We wrote the proof out in LaTeX, adjusting the wording to fit an audience new to notions of divisibility and congruence as we went. Here is the LaTeX source file, and here is the resulting PDF.
The importance of exploring examples (and non-examples) of a new claim or idea to become comfortable with what it means, the definitions it relies on, etc.
Divisibility and congruence.
Review of direct proofs and formal proof writing.
Proof via the contrapositive.
In section 3.2, read...