Introduction

Welcome...

... to Math 239 (Proofs)

I’m Doug Baldwin

Misc

Colloquium this week!

Thursday, Jan. 24, 4:00 - 5:00 PM

Newton 203

“Mathematical Phylogenetics: A Summer Research Possibility”

Dr. Joseph Rusinko, Hobart & William Smith Colleges

Partly a pitch for an REU at Hobart & William Smith, so a good thing to go to if you are looking for ways to do research this summer.

Why Proofs?

People start making surprising claims about math, and you need to be convinced.

For Example...

Put the following sets in order, from the one with the most members to the one with the fewest:

• The natural numbers, i.e., 1, 2, 3, ...
• The positive rational numbers, i.e., 1, 2, 1/2, 1/3, etc.
• The positive real numbers: 1, 2, 0.5, π, etc.

Thoughts:

• Since the reals contain all the rationals plus some irrational numbers, and the rationals contain all the naturals plus some others, clearly there are more reals than rationals and more rationals than naturals.
• Or maybe since all three sets are infinite, they’re really all the same size.

Surprise: the sets of naturals and rationals are the same size, despite intuition to the contrary, while the set of reals is larger than both of them.

Course Preview

We’ll end the semester by understanding why the above surprise is right.

But to get there we’ll have to lay a lot of groundwork, including...

• How do you talk about whether two infinite sets have the same size? You can’t just count their elements like you could for { a, b, c }. We’ll use (and learn about) functions aka mappings aka correspondences to do this.
• We have to have a good understanding of sets, of course.
• We have to have lots of methods for reasoning about mathematical statements — basically, proof techniques.
• And finally, we have to know exactly what a “mathematical statement” is (e.g., does “x = y + 1” count? It certainly sounds mathematical. What about “this statement is false”?)

Learning

How well should you learn the things discussed above?

Here are some different ways in which you can be said to “understand” something, based on something called “Bloom’s taxonomy.” Which do you think will receive the most emphasis in this course? (Stars represent our collective analysis after discussion.)

1. * Memorizing facts, definitions, theorems, etc.
2. * Restating facts, definitions, theorems, etc. in your own words
3. * Using facts, theorems, etc. to solve problems
4. * Explaining/exploring the relationships between different facts, theorems, etc.
5. ** Proving theorems
6. Discovering new extensions to math

What are some things that would help you understand at that level?

• Going to class
• Going to office hours
• Asking questions
• “Doing” it on your own, i.e.,
  • Reviewing notes/book/etc for yourself
  • Practicing solving problems/writing proofs (My experience is that this is among the best ways to learn, and I’ll create lots of opportunities for it in this course.)

Next

Syllabus and Course Policies

Read the syllabus

Mathematical Statements

What makes a statement “mathematical”?

Think about the examples earlier, e.g., “x = y + 1,” “this statement is false,” as context for reading.

Read section 1.1 of the textbook up through “Techniques of Exploration” (beginning of page 1 through first 1/3 of page 5).

The textbook is Sundstrom, Mathematical Reasoning: Writing and Proof (version 2.1), available free online.