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: G_x, G_y, and G_0 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 y = x^2“, 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): G = \{ (x,y) | y = x^2\}. 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”: G_x := \{ (x, -y) | (x,y)\in G\}. 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, {\Bbb R}^2 (“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 G \subset {\Bbb R}^2“.

Nearly done; lunch won’t wait. Some student should now impress me by finding out how to make the symbol {\Bbb R}^2 appear in a comment (don’t forget to use a fake name).

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  1. brutus12

    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.

  2. 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.




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