Math 2710, Fall 2012
You can find a quick cheat sheet on Boolean connectives here. As I said in class, you are encouraged to read Section 1.1: it has some very basic mathematical vocabulary that you'll be using for the rest of your mathematical career.
|Week ||Sections in text (estimated) ||Test ||Administrivia|
|Week 1: Aug. 27-31 ||1.2-1.5 || |
|Week 2: Sept. 3-7 ||1.6, 2.1, 2.2 || ||No class on Monday (Labor Day)|
|Week 3: Sept. 10-14 ||2.2, 2.3 || ||Monday: last day to drop without a W |
or change grading option to pass/fail
|Week 4: Sept. 17-21 ||2.5, 3.1, 3.3 || |
|Week 5: Sept. 24-28 ||3.4-3.6 || |
|Week 6: Oct. 1-5 ||7.4 ||Midterm #1 |
|Week 7: Oct. 8-12 ||4.1, 4.2 || |
|Week 8: Oct. 15-19 ||4.3, 5.1, 5.2 || |
|Week 9: Oct. 22-26 ||real numbers (not in text) ||Midterm #2 |
|Week 10: Oct. 29 - Nov. 2 ||6.1-6.3 || ||Monday: last day to drop |
or change a pass/fail option to letter grade
|Week 11: Nov. 5-9 ||6.4-6.6 || |
|Week 12: Nov. 12-16 ||8.1-8.4 || |
|Week 13: Nov. 19-23 || || ||Thanksgiving break!|
|Week 14: Nov. 26-30 ||8.5-8.8 || |
|Week 15: Dec. 3-7 ||amusing topic and review || |
Be aware that saying "there exists an x" doesn't mean that there's only one, nor does it mean that there can't be only one. It's equally true to say "there exists a prime number greater than 3" (when there are infinitely many of them -- we'll prove that later) and "there exists a prime number less than 3" (when there is exactly one such prime).
There's a nice video about truth tables here. Take a look if you've having trouble with them!
I encourage you to read Section 1.5. I talked about the most important proof techniques in it and you had an in-class exercise on one more, but there are a few others that it wouldn't hurt you to think about before you have to use them.
You should read the proof of Proposition 2.11. Better yet, you should try to prove those statements yourself and then compare your proofs to the book's!
If you want to practice the Extended Euclidean Algorithm, the good news is that you don't need me to give you sample problems. Just pick two integers and get to work! What happens if you let b be bigger than a instead of the other way round? And how would you do this if either a or b were negative?
At this point, you might want to try reading the proof again and seeing if it makes more sense now that we've done some examples in class.
Today's reading assignment is Props. 2.27 and 2.29. We've done examples of the Linear Diophantine Equation Theorem; we'll prove it on Tuesday.
If you'd like to know more about the Hilbert problems, you can find his original address here, and if you'd like to see some of the people involved in the solution to Hilbert's Tenth, you can watch this trailer for a movie about Julia Robinson. Davis, Putnam, and Matiyasevich all appear. (One fact that is not obvious from this video is that Constance Reid was not only a mathematical biographer but also Julia Robinson's sister.)
I said in class today that I was skipping section 2.4. It's a good read if you're interested in converting numbers from one base to another (for instance, if you're a CS major and spend lots of time in binary). The song I mentioned is Tom Lehrer's "New Math," which can be seen lip-synched here. Now, can you figure out why Oct. 31 is the same as Dec. 25?
I proved the theorems about finding gcds and lcms very quickly. I recommend that you read over Prop. 2.56 (this one is the basis for the others that I just stated as a "fact") and Ths. 2.57-59. It's a good exercise in learning to read proofs involving an arbitrary element of a list (that is, looking at each p_i individually).
Today's reading suggestion: take a look at the proof of Prop. 3.12. Better yet, try to prove it yourself first and then see how the authors of the book did it.
The first midterm I gave in 2710 last spring can be found here. Don't worry about #7 -- we haven't gotten to that point yet.
Consider the function f(x,y)=x^2+y^2 on R^2, and define the relation R so that the point (p,q)R(r,s) if f(p,q)=f(r,s). Show that R is an equivalence relation and describe its equivalence classes.
If you want to go more in depth in number theory, you could try proving Wilson's Theorem: if p is prime, then (p-1)! is congruent to -1 modulo p (Chapter 3, #79). Or you could read about the Euler phi function (defined on p. 78 and discussed for a couple more pages). It's fun!
We've just started induction. Keep in mind that the necessary ingredients for a set you can prove things about by induction are (1) some least element that you can use in your base step and (2) a rule that tells you how to get from one element to the next. Normally, we use P, where the least element is 1 and we get from one element to the next by adding 1. However, we could use the positive odd integers, in which case the least element would still be one but you get from one to the next by adding two. This means that for your inductive step, you'd show that if P(k) is true and k is odd, then P(k+2) is true. How would you set up an induction proof if you wanted to prove a statement about every positive integer congruent to 4 modulo 12?
I strongly suggest that you finish the proof that the formula I gave you for the nth Fibonacci number works: it's #63 in Chapter 4 of your textbook.
Here's another induction problem for you: Suppose that you have a string of Hs and Ts generated by putting down a row of coins and looking to see if they're heads or tails: for instance, HTTTHTHTH. Now you can modify this string in a certain way: you're allowed to remove any H and flip the coin(s) on either side of it, but not to close the gap created by removing the H (for instance, HTTTHTHTH could be turned into HTTH.HHTH, where the dot represents the gap, or HTTTHTHH.). You can keep doing this over and over and possibly get rid of all the coins in this way: for instance, HTTHH can become .HTHH, then .HTT., then ..HT., then ...H., and finally ..... at the end. On the other hand, you can't reduce TTT at all! Prove that you can remove all the coins in this way iff the number of heads in the original string is odd.
First of all, you can find the second midterm I gave my Transitions class last spring here. Don't worry about #4, but the rest would be reasonable to ask you.
Here's a carefully-written version of the statement and "proof" I gave you in class today. It will make clear that one possible complaint doesn't lead to a problem.
Statement: All math professors are female.
Proof: Let P(n) be "In any set of n math professors, all of them have the same sex." Our base case is 1: Clearly, in a set of 1 math professor, only one sex will be represented. Now suppose that P(k) is true: in any set of k math professors, all of them have the same sex. We want to show that in any set of k+1 math professors, all of them have the same sex. Suppose you have a set of k+1 math professors. Take a subset of them of size k, leaving out person A. By our induction hypothesis, we know that everyone in that subset has the same sex. Now we remove one person from that set and put person A in instead. Since we have another set of size k, we know that everyone in this set has the same sex. This means that person A has the same sex as everyone else in the set of size k, so everyone in the group of size k+1 has the same sex and the induction hypothesis is proven. And since you know a female math professor, you can see that all math professors are female!
Today we looked at a couple different ways of defining the rationals from the integers. One way gave us a simple definition of the set of rationals but a more complicated way to handle operations, and the other way gives us a more complicated definition of the set but an easier way to define the operations. Both work, but we'll use the second method when we talk about them formally. At this point, though, we can just think about them as the rationals we know and love.
The homework tips you all provided on your homework last week can be found here.
A pretty good explanation of Dedekind cuts can be found here. Before you read it, though, think about how you'd define the sum of two Dedekind cuts as another Dedekind cut!
One of the other comments on those surveys about how I could improve was simply "Vi Hart." I don't know whether this means I should try to be more or less like her, but since it was just Halloween, it seems appropriate to link to a video of hers about eating candy buttons today.
The definition of convergence of a sequence begins with "for every positive epsilon, there is a positive integer N." While I couldn't find a song about that, this song by Tom Lehrer concerns the epsilon-delta definition of the continuity of a function. Enjoy!
Think about what it means to be a function. Try defining functions on sets you normally wouldn't use: the students in the Math 2710 class, the snowflakes that have fallen on the UConn campus in the past 48 hours, or the books on your bookshelf.
As a bonus, you can think about the function I defined today that is 1 at every rational number and the identity at every irrational number. Come see me if you'd like to talk about what it means for a function to be continuous.
First of all, I'd promised you a resource for the analysis part of the course. There's a nice set of lecture notes here. You'll be most interested in Chapter 3.
Second of all, you can test your understanding of injectivity and surjectivity by defining functions from, for instance, the negative integers to the positive integers that have each combination of those properties.
I found a video on Hilbert's hotel! If you'd like a less animated treatment of infinite cardinalities, this New York Times article is a good one.
Properties (ii)-(x) of Prop. 8.25 show that the complex numbers form a field: addition and multiplication are both commutative and associative, 0 and 1 are part of the complex numbers, there are multiplicative and additive inverses, and the distributive law holds. The reals, the rationals, and Z_p for p prime also form fields. Which of these required properties does P not have? What about N, Z, and Z_m for nonprime m?
The propositions I stated but didn't prove are Prop. 8.42 and Prop. 8.43. I suggest looking at them to make sure you understand how they're done, especially the proof of the triangle inequality (Prop. 8.44(iv)), which is by far the most complicated.
I think this is what you'll all want most: last spring's final exam. Don't worry about 1(e) or 9: that material is from Chapter 9, which we didn't cover (I opted to do more work with sequences of real numbers instead).
To understand the geometric interpretation of taking nth roots of complex numbers a little bit better, choose a manageable n and find the nth roots of 8 cis (pi/4), 8 cis (2pi/3), and 1 cis (2pi/3). Then plot them. How do the plots differ from each other? Why?
If you want to understand the proof of the Fundamental Theorem of Algebra better, I suggest that you start by taking an equation like
2z^4 + 3iz^3 + (1+2i)z^2 + z - 4i = 0
and work through the proof with this particular equation.
Good luck to all of you as you study for your finals. It's been fun teaching you!