Monthly Archives: January 2009

Algebra and the definition of continuity

I spent most of yesterday working on the algebra homework that was due today and once again I have to say the assignment was really well put together. The problems were mostly about showing something is a group or a subgroup, but the concept of isomorphism came up a couple of times as well. All and all I think I learned quite a bit from doing this assignment.
One of the problems required that you show that certain functions were continuous. Personally I thought this was one of the easier problems that was assigned, but near the end of the homework session yesterday I was helping a couple people with it and I told them that the proof was almost a trivial application of the definition of continuous. They both looked at me like I had two heads. Now neither of them has studied much analysis beyond the required calculus sequence and now that I think about it I can’t really remember how continuous was defined in my first calculus class.
I know we discussed continuity since I remember learning the intermediate value theorem and obviously there needs to be some understanding of continuity to discuss derivatives, but even though it has been awhile I don’t think the delta-epsilon definition of continuity was ever mentioned. So how was continuity defined?
I tried to consult my old calculus text, but I’m not exactly sure where it is at the moment. So I did a search trying to find out how AP calculus classes were defining continuity and the only thing I found was defining continuity at a point if the limit as x goes to c exists and is equal to f(c). Which is all fine and dandy, but the intermediate value theorem requires continuity over an interval. So how was continuity over an interval defined in my calculus class? I suppose someday I will find my old text and have an answer.

N+1 pigeons and only N pigeon holes

I spent most of the day yesterday working on the combinatorics homework that was due today. All of the exercises required the application of the pigeon-hole principle, but well I suppose that is to be expected after we spent the last week or so talking about the pigeon-hole principle in class. I am certainly glad that we are starting another section where not everything will have been contrived so as to make it possible to use the pigeon-hole principle. Anyway a couple of the exercises on this past homework were kind of interesting so I thought I might write about them.

In a room of 10 people, each of which has an age (in whole years) of at least 1 and not more than 60, show that you can select 2 groups with no common person such that the sum of the ages of the people in the 2 groups are equal. This problem took me a while to figure out, mostly because I was stupid and tried to construct disjoint subsets. Once I realized that I didn’t need to construct disjoint subsets since if I had 2 subsets with common elements, but equal sums I could just take out the common elements to get disjoint subsets with equal sums the problem was pretty easy.

The other problem I found interesting was probably just because it was geometric in nature. Basically there was a square with side length 2 and we had to show that if you placed 5 points within the square than the distance between them would be at most \sqrt{2} . The basic idea is that if you center the square on the origin than you can cover the entire square with circles with diameter of \sqrt{2} centered at (\pm 1/2, \pm 1/2). Now if 2 points are in the same circle than the distance between them is at most \sqrt{2} and there are 4 circles and 5 points.

Algebra

Today we had a study group meeting for my algebra class to work on the homework that is due tomorrow. I have to admit that I was apprehensive going into this since I have always shied away from studying in groups, but I have decided to give it a try at least some just to see how it works. I was pleasantly surprised that I seemed to gain a lot from being there; even though I was often in a “teacher” type of role I found that trying to explain my ideas to other people often times forced me to state them in different more clear ways.

The one problem that seemed to excite the most discussion was a problem that defined two equivalence relations on R^3 such that P and P’ are related if they are planes in R^3 that have the same normal vector and L and L’ are related if they are lines in R^3 that have proportional direction vectors. Essentially, the relations defined equivalence classes which consisted of parallel planes and lines. In fact I think this was the definition given in the statement of the problem, but I found it easier to look at the classes as sets of lines and planes defined by the same vector.

Anyway the first part seemed almost trivial. All it asked was that given a plane P in R^3 to show that there was a unique plane p_0 passing through the origin such that p_0 is an element of [P] and then to show that given a line L in R^3 there was a unique line l_0 passing through the origin such that l_0 is an element of [L].

Moving on to part b is where things got much more complicated. RP^2 was defined to be a 2-dimensional geometry whose points are equivalence classes [L] and whose lines are equivalence classes [P]. A point [L] is incident with a line [P] iff l_0 is a line in the plane p_0 . Part B asked for a proof that any two points of RP^2 lie on a unique line of RP^2 . This turned out to be not so bad once I was able to get my head around what it was asking. Pretty much it is asking you to prove that you can define a plane by two distinct lines that lie within the plane.

Part c then asked for a proof that any two lines of RP^2 intersect at a unique point in RP^2 . Most of the discussion here again took off in the direction of planes in lines in R^3 , but a couple of us found a different approach that almost seems like cheating the problem. Simply assume that there are 2 distinct lines [P] and [P’] in RP^2 that don’t intersect at a unique point. Now either [P] and [P’] don’t intersect in RP^2 or [P] and [P’] intersect at more than one point in RP^2 . But in the case of the former this implies that all elements of [P] don’t intersect any elements of [P’] which can’t happen since [P] and [P’] are distinct sets of parallel planes. In the latter case you can quickly contradict the result from part b since you now can define two distinct points in RP^2 that both lie on two distinct lines in RP^2 . Thus any two lines in RP^2 must intersect at a unique point.

Addition and multiplication

One of the problems for my algebra homework was to show that addition and multiplication are well defined on Z mod n. The addition part was pretty easy just let [a]=[a’] and [b]=[b’] and now everything pretty much follows from the definitions. a-a’=kn and b-b’=mn for some integers k and m. Now (a+b)-(a’+b’) gives some terms divisible by n and then (a-a’) and (b-b’) terms which of course are also divisible by n.
That didn’t take much time at all to figure out. Trying the same thing on multiplication didn’t seem to work out very well. (It may very well work, but I failed to accomplish anything more than making a mess.) Sadly it took me a little while to stumble on to the trick of using the division algorithm to represent a, a’, b, b’ and showing that the remainders for the congruent variables were the same. From that point it was easy to show that ab-a’b’ was divisible by n since the remainders canceled and I was left with a bunch of terms that contained n and thus were divisible by n.

Winter Quarter

Today was the third day of classes so far this quarter. I am taking five classes this quarter: Analysis, Algebra, Combinatorics, Stats, and Anthropology. Of course I wanted to take Italian, but it wasn’t to be, the waitlist wasn’t shrinking and they didn’t open another section so I ended up in Combinatorics. This should turn out to be a fun quarter.
Algebra seems really cool. Most of this quarter we are going to end up focusing on group theory, but so far we have just rehashed some set theory from the foundations class I took last quarter and defined equivalence relations and equivalence classes. The first day of class he said that the homework was going to be pretty difficult and typically that means that it will be interesting. We got the first homework assignment today, but I haven’t had a chance to even look at it yet so I can’t really comment on it. It is due on Wednesday, but the professor is holding an additional class meeting on Tuesdays to work on homework so I want to have most of it figured out by then.
Combinatorics is a hard class to get a grasp on. So far there have just been some motivating problems designed to get us to see the types of problems that combinatorics can solve, although today we did start looking at the pigeon-hole principle and how it can be applied to various combinatorial problems. The first homework is due tomorrow and I have it mostly done, there were a couple of interesting problems on there that I might comment on sometime in the future.
Stats is well stats. I’m still not sure that I am excited about the subject, but I think this quarter probably will be better than the previous. To begin with the lectures are less boring. They still more at a slower pace than I am accustomed to, but at least I don’t have to fight to stay awake. Add to that the homework appears to be if not more interesting than the homework last quarter at least shorter and less repetitive.
Analysis has been pretty annoying so far. The class has met twice and both times it has been pretty much a repeat of foundations from last quarter. The first class was mostly just induction and today it was cardinality of sets, including repeat proofs of at least 4 results from last quarter. Hopefully Friday’s class will introduce some new material since I think he is running out of stuff from last quarter to prove.
My Anthropology class has only met once thus far, but looks like it is going to have a fair amount of reading required, but other than that it doesn’t seem like there is much work involved. It was kind of funny that when I arrived to the class, probably about 20 minutes early, there were about 15 students standing outside of the empty classroom. Once I went in and sat down they all came in and sat down as well. I’m not sure if they were unable to figure out how to open the door or if they finally figured out that sitting down was probably preferable to standing. Either way it kind of scares me to be in a class with these people.
On Tuesday I went to the Reading Classics seminar. I think I am going to try and attend all of them, but I don’t think I am going to formally register for it since I don’t want to commit to anything yet, but given that one of the organizers was already suggesting topics to everyone in the room my guess is that I will probably be badgered into presenting something sometime this quarter. We are looking at Lagrange and his contemporaries and the topic yesterday and spilling over into next Tuesday was Lagrange and the three-body problem. I’ve never really been interested in this type of problem, but I do have to say is fairly interesting to see how people would go about attacking a practical problem such as this.