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

January 6, 2009 in Math 148

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 ( $\mapsto$ ): to reflect-in-the-x-axis, for example, we’re applying the transformation $\null [ (x,y)\mapsto(x,-y)]$ . What I didn’t do is give it a name … which I hereby do. Let $\rho_x$ denote “reflection (of the plane ${\Bbb R}^2$ ) in the x-axis”: $\rho_x := [ (x,y)\mapsto(x,-y)]$ (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
$id:= [(x,y)\mapsto(x,y)]$

$\rho_x := [ (x,y)\mapsto(x,-y)]$
$\rho_y := [ (x,y)\mapsto(-x,y)]$
$\rho_0 := [ (x,y)\mapsto(-x,-y)]$
$\rho_{\iota} := [ (x,y)\mapsto(y,x)]$

$r_n := [ (x,y)\mapsto(x+n,y)]$
$l_n := [ (x,y)\mapsto(x-n,y)]$
$d_n := [ (x,y)\mapsto(x,y-n)]$
$u_n := [ (x,y)\mapsto(x,y+n)]$

$\sigma^x_n := [ (x,y)\mapsto(nx,y)]$
$\sigma^y_n := [ (x,y)\mapsto(x, ny)]$ .

The subscript of $\rho_{\iota}$ is an “iota” (Greek “i”) and here stands for the Identity Function $\iota(x) = x$ ; 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 $\sigma^x_n$ 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 $\sigma^y_n$ .

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 $\rho_0 = \rho_y\circ\rho_x$ . We could even go on to mention that $\{ id, \rho_x , \rho_y, \rho_0\}$ , 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 $\null {\Bbb R}^2$ explicitly from day one) is that (“Let G be a Graph”) if we put $G := \{ (x, f(x) ) \}_{x\in{\Bbb R}}$ (so that G denotes “the graph of f“), we can compute as follows:
$l_n(G) = \{ (x-n, f(x) ) \}_{x\in{\Bbb R}} = \{ (q, f(q+n) ) \}_{q\in{\Bbb R}}$ —
the “reason” that “adding a positive number ‘inside’ the function” is associated with a shift to the left (somewhat counter-intuitively).

4 Comments

brutus12

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!!

Reply
vlorbik

January 7, 2009 at 12:24 pm

hey, the system works!
thanks, brutus12!

Reply

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