### {(0,0),(0,1),(1,0),(1,1)}

OK, no comments from my new students. So be it. Maybe I’ll offer a bribe (it’d have to be in the form of, say, a puzzle book … class credit is right out of the question). Anyhow, the PM class went OK … don’t know if we got two-and-a-half *days* worth done …

What follows will mostly be too math-y for actual 148.

We’ve already used the “maps to” symbol (): to reflect-in-the-*x*-axis, for example, we’re applying the transformation . What I *didn’t* do is *give it a name* … which I hereby do. Let denote “reflection (of the plane ) in the x-axis”: (that’s a “rho”—a Greek “r”). While we’re at it, let *n* denote an arbitrary nonzero real number. Here are some more**Definitions**

.

The subscript of is an “iota” (Greek “i”) and here stands for the Identity Function ; reflection in the graph of this function (as we will see) replaces the graph of a relation with the graph of its *inverse* relation. Also, the Greek part of the symbol is a “sigma” (Greek “s”); the whole symbol stands for “Scale the *x* variable by a factor of *n*” (*i.e.*, [as the code indicates], multiply each *x* co-ordinate by n) … *mutatis mutandis* for .

If we now only knew about stuff like Composite Functions, we’d be able to develop quite a bit of pretty interesting theory. I’ve already observed, for example, that “reflection in the origin” can be computed by first “reflecting in *x*“, then “reflecting in *y*“; with our new symbols this becomes . We could even go on to mention that , with the operation of composition, form a (so-called) Group (specifically, one in the isomorphism class called The Klein 4-Group [no, not the guys in this video]). Stuff like that. Some of it’ll actually come up in class when the tools fall into our hands.

In the meantime, what’s *really* got me excited about this whole program (work with Transformations on explicitly from day one) is that (“Let G be a Graph”) if we put (so that *G* denotes “the graph of *f*“), we can compute as follows:

—

the “reason” that “adding a positive number ‘inside’ the function” is associated with a shift to the *left* (somewhat counter-intuitively).

January 7, 2009 at 1:41 am

I like that what we go over in class is on here. I struggle with math, so being able to come here to go over what we did in class is awesome!! Hats off to Vlorbik!!

January 7, 2009 at 12:24 pm

hey, the system works!

thanks, brutus12!