Math 2710-001, Spring 2014
I've made up a summary of the connectives we talked about in class today (and a few we haven't gotten to yet!). Take a good look at the "if...then..." section.
|Week ||Sections in text (estimated) ||Test ||Administrivia|
|Week 1: Jan. 20-24 ||1.2-1.5 || ||No class Monday|
|Week 2: Jan. 27-31 ||1.6, 2.1, 2.2 || |
|Week 3: Feb. 3-7 ||2.2, 2.3
|| ||Monday: last day to drop without a "W" |
or choose the P/F option
|Week 4: Feb. 10-14 ||2.5, 3.1, 3.3
|Week 5: Feb. 17-21 ||3.3-3.5
|Week 6: Feb. 24-28 ||4.1
||Midterm #1 |
|Week 7: March 3-7 ||4.2, 4.3
|Week 8: March 10-14 ||5.1, 5.2, real numbers (not in text)
|Week 9: March 17-21 ||
|| ||Spring break!|
|Week 10: March 24-28||real numbers (not in text)
||Midterm #2 |
|Week 11: March 31- April 4 ||6.1-6.3
|| ||Monday: last day to drop |
or choose to get a letter grade
|Week 12: April 7-11 ||6.4-6.6
|Week 13: April 14-18 ||8.1-8.4
|Week 14: April 21-25 ||8.5-8.7
|Week 15: April 28 - May 2 ||8.8 and review
Thursday, January 23
Two important things to keep in mind from today:
Your recommended reading for the day is Proof Methods 1.56-1.59 in Section 1.5. If you understand the general ideas behind them, your semester will be a little bit easier.
- If you negate a quantifier, you "pull the negation through" and change the quantifier. If you negate (A and B) or you negate (A or B), you "pull the negation through" and change the "and" to an "or" or vice versa.
- Keep in mind that order of quantifiers matters a lot! (exists y)(for all x)L(x,y) is different from (for all x)(exists y)L(x,y). Does one of these imply the other?
Thursday, January 30
If you want to finish reading the proof of the Division Algorithm, take a look at 2.12 in Section 2.1. You might also consider a way to adapt the Division Algorithm for negative b: how could you change the statement to make it true in that case?
Take a look at the proof of the GCD Characterization Theorem (2.24) and work through it on your own! If you have questions, please come ask me.
Thursday, February 6
The most important thing you can do to get used to the Extended Euclidean Algorithm is to practice it. Try problems 19-26 in Chapter 2 if you don't want to make up your own numbers.
Your reading suggestion for the day is Props. 2.27 and 2.28. Enjoy, and let me know if they don't make sense!
Now you know how to find out if a linear Diophantine equation has a solution and what the solutions are. Go through some of the examples to see how a word problem involving such an equation might be written.
Thursday, February 13
Your reading assignment for the day is Theorem 2.53.
If you want a song about math in different bases (Section 2.4, the one we skipped), listen to this Tom Lehrer song.
Thursday, February 20
The last 2710 midterm I gave can be found here. You wouldn't be responsible for the material in questions 6 and 7 (you can tell we didn't have snow days two falls ago!).
Today's reading assignment is Prop. 3.12.
At the end of class, we found out that not every equivalence class modulo 4 has a multiplicative inverse: there is no n such that  times [n] equal . In preparation for Tuesday's class, I suggest that you work out the times tables for arithmetic modulo 5: does every equivalence class modulo 5 have an inverse?
If you thought Fermat's Little Theorem was neat, take a look at Wilson's Theorem (#79, Problem Set 3).
Thursday, March 6
While I'm skipping the rest of Chapter 3 and Chapter 7 (the cryptography chapter), you have almost enough background now to understand why RSA works. You just need the Chinese Remainder Theorem (3.62)!
Here is a statement that can be proven by induction. If you turn in the proof with your homework on Thursday, I'll factor the points you earn into your homework assignment as extra credit.
For all positive integers n, 6 divides (2n^3 + 3n^2 + n).
Thursday, March 13
Here is last year's second midterm. I expect you to be able to do all the problems on it except #5 as well as problems #6 and #7 from last year's first midterm.
Look at the proof that the square root of two is not rational. Work out a similar proof that the square root of 6 is not rational. Try to use the same technique to prove that the square root of 4 is not rational. Where does it fail?
You can find a good set of analysis lecture notes here. The most important chapter for you will probably be Chapter 3.
Thursday, April 3
We talked about proving that sequences converge and diverge today. Remember the order of the quantifiers: to show that a sequence converges, you need to
To show that a sequence diverges, you need to
- take some arbitrary epsilon,
- find an N that guarantees |p_n - p| < epsilon for all n greater than N, and
- prove that this N works.
Try it yourself with the sequence p_n = 1/n^2 for positive integers n.
There are two main things you should remember about functions at this point. The first is that a formula isn't necessary to define a function, but a rule is: there must be a clear way to define f(x) given x. That way can be a formula but doesn't have to be. The second is that the definition of a function is not complete without specifying the domain and codomain.
- find an epsilon such that
- no matter what N you choose,
- you'll always be able to find some n>= N such that |p_n - p| >= epsilon.
Thursday, April 10
There are some things I'd like you to keep in mind when you write proofs by induction in the future:
Try proving that the function I gave as a bijection from P to PxP really is one, and then try to work through a few more details in the sketch of my proof that #P=#Q.
- State P(n) explicitly. Note that P(n) is a statement about a single n, so don't add "for all n" at the end of it.
- State explicitly where you use the induction hypothesis.
- At the end, say something like "Since we've shown P(1) is true and that for all k, P(k+1) is true if P(k) is true, P(n) must be true for all positive integers n."
- Don't write something is true before you prove it.
Thursday, April 17
You've seen that there is a bijection between pairs of real numbers and the complex numbers. This means that #RxR=#C. Is the cardinality of #RxR greater than the cardinality of R?
Your reading assignment for the day is Prop. 8.42. If you're not comfortable doing arithmetic with complex numbers, please practice a little on your own.
Thursday, April 24
Today's reading assignment is Prop. 8.44. Reading Ex. 8.45 might also be illuminating.
This really goes back to last week, but here is a funny little video about Hilbert's Hotel.
Please bring your notes to class on Thursday: we'll be finishing the proof of the Fundamental Theorem of Algebra! If you weren't in class today, please get a classmate's notes before Thursday.
Here's the final exam from the last time I taught 2710.