### Discrete Math Final: Questions 3, 4, 5, 6, and 7

Very close… embarrassingly close… to the questions of the recent quiz. The “A” students obviously get pretty bored seeing the same thing over and over. And I’ll freely admit that I’ve been more or less ruined as an instructor for these (reasonably well-prepared) students by (what seems like) a lifetime of “remediating” the losers of the system. If you’re gonna be any good at preparing the well-prepared (or so it seems to me right now), you’re gonna have to embrace the “blame the victim” strategy: “we’re going at this pace and it’s your fault if you don’t (‘want to’) keep up”.

Meanwhile, what every (all-too-human, somewhere-in-the-middle) teacher actually does in practice is compromise.

In an ideal world, we’d do “so-and-so”. Certain smile-all-the-time manipulative assholes will therefore pretend that if “so-and-so” doesn’t happen, you’re to blame (and’ll have to be punished severely). It’ll behoove you, then, if you can tell which sides of the cards have the spots on ’em, to pretend that this system makes sense and file certain paperwork alleging to show that so-and-so has been done. “Be realistic.”

Is a syllabus a “contract”? Not in my world. But then, in my world, contracts rightly-so-called can only be entered into by parties of approximately equal power. To put this as forcefully as I know how… though I don’t expect you to get it… if corporations are “persons” in The Law, then actual living beings are something else… and this is obvious. These entities just want to get us on the record admitting that everything is our fault so they can use it against it at their whim. And this is obvious.

Still. Denying the holy spirit is the unforgivable sin; I’m damned if I’m gonna give up hope. This class… as I’ve hinted… came to me wanting to find out more about how math-heads go about their business and willing to work hard to find that out. This was the closest thing to… well… whatever it was I’ve been working so hard all my life to get ready to do… as I’ve got any reasonable hope ever of attaining. It was a blast.

Naturally, most of ’em were, in some sense, “lazy”. One has a lot of commitments; if certain assignments don’t “count”… or remain unassigned altogether… well, then, time is short. Meanwhile, I am committed to the principle that “no ‘serious’ student fails”. (For some hugely subjective value of “serious”.)

So here’s how it works out. I put a bunch of questions on the quizzes and tests that properly belong on some much lower-level tests according to the “hardliner” this-is-college-kid model. And then I’m very generous with “partial credit”. Anybody who’s still failing is really failing… and I’ll typically have a pretty easy time getting ’em to admit this when we talk about it face-to-face. Meanwhile, a lot of on-somebody-else’s-model “undeserving” students are getting Gentleman’s C’s.

The upside… enormous if I took my “math educator” role seriously… is that I encounter actual data about how students of whatever material is at hand actually encounter the given material. The signal-to-noise ratio on this particular subject is very nearly zero in my experience (libraries and the net), so, anyway for me, there’s a great deal to be learned from (carefully) marking papers written by these starting-to-become-advanced students… particularly when the material is of the usually-taken-for-granted variety treated here.

3. Let $U= \{0, 1, 2, 3, 4, 5, 6\}$ be the Universal Set (for this problem) and let $X=\{0,1,2,3\}$, $Y=\{0, 3, 6\}$, and $Z=\{0, 5\}$.

Compute the given sets. (Remark: we will have shown that set difference is not associative).

$(X-Y)-Z}$
$X-(Y-Z)$
$Y^c \cup Z^c$
$X \cap X^c$

at this point wordpress decided to fuck me with a redhot poker. one gets bored. some other time maybe.

{\bf 4.} Let $X = \{0,1,2,3\}$ again; compute $\wp(X)$ (the power set of $X$).
\vfil
{\bf 5.} Let $Y = \{0,3,6\}$ and $Z = \{0,5\}$ again;
compute {\bf a.} $Y\times Z$ and {\bf b.} $Z \times Y$.

(We will have shown
that “forming the Cartesian product” is not commutative.)
\vfil\eject
{\bf 6.} Let $U = \{0, 1, 2, 3, 4, 5, 6\}$ again.
Write out the Relation $R$ defined on $U$ by $xRy \equiv 3 | (x – y)$.
\vfil

{\bf 7.}
Let $f(x) = log_3(x)$, $g(x) = 5x+1$. Find (“formulas” for) the following functions.

\line{
{$f^{-1}$}\hfil
{$g^{-1}$}\hfil
{$f\circ g$}\hfil
{$g \circ f$}
}
\vfil