Archive for the ‘Sets’ Category

In Media’s Rays

Posted by vlorbik on November 14, 2009

Finite Sets are most easily displayed
by listing their elements (and, when it’s
convenient so to do, also naming them):

A = {a, b} and T = {a, e, i, o, u}

for example.

We note that A\=a here;
this means that
$A \not= a$
and that typing math
is here considered as
part of the problem
of the solution.

[
The point of having so noted
is that we are, as we so often do,
pretending to begin again.
This is in part the effect of
but in this case is also something
of a conscious choice. Anyhow,
I’ve been going on about handwriting
for a while now and intend to
continue. Hypertext is all well
and good and I’ll drop links
as usual according to my whim
(or careful design… you never
can tell… [until you can]… but
there’s nothing like the body
of the post
when you want
to something).
]

Make sure that whatever symbols
you use to represent my “A” and my “a”
are easily distinguished. This kind of thing
can sometimes be more trouble than you’d
think. For example,
something of a problem for me,
not in writing down credible versions
of
a\=A
b\=b
or
d\=D
but in… but you’ll have guessed…
c\=C
o\=O
and suchlike U&lc
(upper-and-lower-case [RIP])
pairs distinguishable,
as i tend to write them
(unless i’m being very careful),
only by their relative sizes.

Gedanken Experiment (I).
Think through which letterpairs
will give you trouble
(when you’re not careful).

[
in “script” versions of handwritten
letters there are common flourishes
to distinguish, say, script-c
from script-C; “print” letters
aren’t always so easy.

story making very little sense here
(you have to see the letters as i draw them).
my script-y’s and script-z’s
looked too much alike…
and, as you can imagine,
$x$, $y$, and $z$ come up
*a lot* when you stand up
in front of basic-algebra classes…
so i started writing my z’s
*crossing* ’em to tell ’em
from 2’s.
no wait. actually this story
makes *perfect* sense.
it gives me an opportunity
to report that students
won’t believe that this
is a good idea when told
or shown or even when
they themselves make
a the mistake you’ve
work submitted to you
(yourself; that would
be me in this case).
i’m only *vividly* aware
of this having happened
*once* but i can’t swear
it didn’t happen before
or since then. nobody
believes anything you
tell ’em in this sorry
racket and it’s heart-
breaking. heartbreaking
i tell you.
]

Recall that
A = {a, b} and T = {a, e, i, o, u}
.

The set product (or cross product)
of a given pair of sets is the set of all possible
ordered pairs consisting
of a first (or left-hand) entry
taken from the first set and
a second (right-hand) entry
from the second set. Thus

A x T=
{
(a, a), (a, e), (a, i), (a, o), (a, u),
(b, a), (b, e), (b, i), (b,o), (b, u)
},

and

AxA={(a,a),(a,b),(b,a),(b,b)}.

The “carriage returns” in our display
of AxT are here as a convenience;
it’s just as correct… and in some circum-
stances correcter… to smash ’em.

AxT={(a,a),(a,e),(a,i),(a,o),(a,u),(b,a),(b,e),(b,i),(b,o),(b,u)}.

[
i’ve smashed space too.
this can be well worth it.
a *lot* of trouble lies in those
invisible characters.
and if you can learn to read
code that tight–and prove
it by *writing* code that tight–
you’re way beyond this lesson
and it remains only to endure
my plea that you comment on it.
]

Anyhow. This construction
can be taught to beginners
knowing nothing of set theory
in a single lecture obviously.

And without it, the whole
god-damn “functions as sets
of ordered pairs” thing…
the “let’s cram hundreds of years
of post-scientific-revolution
philosophy-of-mathematics
down their tender little throats
in one big enormous blob
of sticky incomprehension” thing…
is doomed. Because what the
hell is any of this going to have
to do with so-called “graphs”
in the student’s imagination
if the instructors haven’t themselves
ever connected these dots?

Well, quite a bit actually when the student
encounter the nonetheless-incredibly-mangled
presentations we get in the “standard” treatments…
I was ahead of the instructor sometimes myself
in early days and it did me a world of good.

But the real point.
and then again and again until everybody can teach it
Because it’s the kind of down-to-the-ground
nobody-will-ever-be-righter-than-*i*-am
this-is-the-*one*-thing-i-know-for-sure-so-far
moral fucking certainty
without which math
is just more mental masturbation.

This is sadly lacking in our culture at large;
obviously there’s only so much we can do
in our classrooms about it. And creating
*programs* for the culture at large is a
fool’s errand or showbusiness or politics
not math. But dammit. Give the poor devils
a chance to be right when the teacher’s wrong
and know it in this one arena I beg of you.
It won’t make ’em into little
subversives-like-Kibrolv
and even if it did, they’ll
be just as easily marginalized
as I’ve been fear not of that.

The trick has been to get ’em into “statistics”
with no probability. But just to talk about rolling
two dice, any math-head worth their salt
requires a cross-product (DxD
where D= {1, 2, 3, 4, 5, 6}).

The display alone
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 45
51 52 53 54 55 56
61 62 63 64 65 66
and a few hours with
a competent fellow-student
to a beginning student
than a wilderness of stix classes
as I see it. So. How to keep
students apart? Online classes baby.

Exercise in case you missed the whole rant.
Let $\eta = \{0,1\}$ and $\zeta =\{\pi, \sqrt2, e\}$.
Write out
(including every bracket, brace, paren,
and comma [and no extras]…
but with space on the page
left to your taste and discretion)
copies of $\eta \times \zeta$ and $\zeta \times \zeta$.

Posted in Handwriting, Notations, Rambles, Rants, Sets | 1 Comment »

there goes the weekend

Posted by vlorbik on November 13, 2009

JD ($1000 ln^{-1}(1)$) notices “codomain”. this looks real interesting.

the party betrays its base. again. (gotta give ’em hope. gotta betray their hope. this ain’t beanbag, kid.)