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 such that P and P’ are related if they are planes in that have the same normal vector and L and L’ are related if they are lines in 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 to show that there was a unique plane passing through the origin such that is an element of [P] and then to show that given a line L in there was a unique line passing through the origin such that is an element of [L].

Moving on to part b is where things got much more complicated. 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 is a line in the plane . Part B asked for a proof that any two points of lie on a unique line of . 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 intersect at a unique point in . Most of the discussion here again took off in the direction of planes in lines in , 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 that don’t intersect at a unique point. Now either [P] and [P’] don’t intersect in or [P] and [P’] intersect at more than one point in . 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 that both lie on two distinct lines in . Thus any two lines in must intersect at a unique point.