### Let G Be A Graph …

I’ve just come from the first meetings of my five-day-a-week classes; fifty minutes is a mighty short hour. I’ll soon meet the two-day-a-week version and have a little more wiggle room. After some preliminary remarks about syllabi and such, “Let G be a graph …”.

We began by considering *reflections* of graphs. I’ve introduced some notations here: , , and stand for the reflections of G in the x-axis, the y-axis, and the origin (respectively). These are easily illustrated at the board (so I did it) but much tougher here in the blog (so I won’t). But to see what’s going on *algebraically*, we’ll have to delve into some technical issues. What’s a *graph* (in this context)?

We’re “really” talking about *relations*. The “graph of the equation “, for example, can be considered as a certain curve sketeched in a co-ordinate frame (as I did in class) or as a certain *set of ordered pairs* (as I haven’t [yet] done explicitly; the two-hour version of the lecture this afternoon will probably get more details here): . One virtue of this point of view is that we can be perfectly specific about what we mean by, say, “the reflection of G in the x-axis”: . The downside is, of course, that *notations are scary*. I’ve ranted here before about the tendency in contemporary textbooks to *avoid* using proper notations; I’m expecting this to become something of a theme in the course.

*Anything* that you have to keep writing out over and over eventually *begs* to be abbreviated somehow; developing a taste for *good* abbreviations is part of Mathematical Maturity. Some of our notations will be used only for a single problem and then thrown away; others will last for a single lecture, a week, or a Quarter; others are so useful that they’ll last hundreds of years (the “=” sign, for example). “Algebra in Plain English”—*i.e.*, doing algebra *without* special notations, isn’t actually *impossible*. Just seriously misguided. According to this History of Algebra, the “rhetorical stage” lasted until the 16th Century;

note that *very little progress* was made in Algebra during this stage (though, for example, Geometry had already been very thoroughly developed for thousands of years before this).

Anyhow, what I *haven’t* yet put into the notes, but plan to tomorrow (or this afternoon): the “right name” for “the (x,y)-plane” in *handwritten* work is, of course, (“Real two-dimensional space” when we’re being formal; “arr-two” when we’re not [or, of course, “the (x,y)-plane” …]). In this context, we can be precise about “Let G be graph …”: we “really” mean “Let “.

Nearly done; lunch won’t wait. Some student should now impress me by finding out how to make the symbol appear in a comment (don’t forget to use a fake name).

January 7, 2009 at 1:36 am

I agree, 50 minutes doesn’t seem long enough. It seems that as soon as I get a grasp on the concept (that usually takes awhile) its time to go but not meeting everyday would make it harder to learn, for me atleast.

January 12, 2009 at 1:42 am

I am enjoying your recent posts (as well as refreshing my own knowledge). Notation is scary and my 9th grade students have amazed me with their capacity to understand it.