Misc
Google Calendar Demo
Math Learning Center Opens Today
M - F mid-day plus M - R evenings; see https://www.geneseo.edu/math/mlc for details
Research Weekend
Friday - Saturday, January 27 - 28
Chance to get a taste of what mathematical research is like.
Prof. Aguilar on Google’s Page Rank algorithm, a key part of how Google decides what pages to list first when you do a search, and being a neat application of linear algebra.
Apply by Wednesday. Should have taken Math 230 and 233, but if not and interested, talk to Profs. Aguilar or Bilgic.
Pre-weekend colloquium (open to everyone) Friday January 27, 3:00 PM, Newton 203
Extra credit for attending colloquium and writing me a roughly 1-paragraph summary of some connection(s) you made between the talk and your own interests, experience, ideas, etc. Only “due date” is that you give me the summary before our final exam in May.
Questions?
What is a “backwards question”?
A question that guides you in the working-back-from-result steps of Sundstrom’s “know-show” tables
Proofs
Section 1.2
Conjectures are usually conditional statements
Prover needs to identify givens / hypothesis, and conclusion
Know-show tables can help identify reasoning that leads from hypothesis to conclusion
You can often use definitions to devise steps in a proof
Example (Progress Check 1.9, question 2): Prove that if x is an even integer and y is an odd integer, then x + y is odd
Ideas in a know-show table, although the ideas came out in a straight flow from hypothesis (not explicitly stated) to conclusion, so there aren’t any backwards steps
| Step | Know | Reason |
|---|---|---|
| P1 | x = 2a, integer a | Definition of even |
| P2 | y = 2b+1, integer b | Definition of odd |
| P3 | x+y = 2a + 2b + 1 | Algebra |
| P4 | = 2(a+b) + 1 | Algebra |
| P5 | a+b = c is an integer | Integers closed under addition |
| P6 | x+y = 2c+1 | Substitute c into P4 |
| Q | x+y is odd | Definition of odd |
Note that thinking of even and odd numbers as being numbers of the form 2a and 2b+1 for integers a and b makes it easier to do algebra based on knowing that something is even or odd than do the colloquial definitions as divisible by 2 and not divisible by 2.
The above table is a nice way of capturing the thinking in a proof, but it needs to be written in prose form to count as a formal proof.
Next
Writing proofs formally
Watch video at https://www.geneseo.edu/proofspace/Ch1Sec3