MEdZ 0.4 debut

May 13, 2011 in Lectures Without Words, Math 151, MathEdZine

at long last. this has been sitting
on the paperpile very-nearly-finished
for quite a while.

i started the “lectures without words”
series early on with 0.1: .
whose cover more or less announced
implicitly that it was one of a series
called . and that
was, like, five quarters ago.
and they’re only 8 micro-size pages.

a couple days ago i inked the graphs
and the corresponding code (the stuff
under the dotted line had *been* inked
and the whole rest of the issue was
entirely assembled). and zapped it off.

and yesterday i passed ’em around
at the end of class (to surprisingly few
students given that i’ve got freshly-
-graded exams). it went okay.

i did right away
(if i recall correctly), and in
high-art style, too (i used a
brush instead of a sharpie).
wasn’t much later.
i have plenty of notes for ,
too, and could knock out a version
on any day here at the studio
(given a couple hours and some
peace of mind) that’d fit right in.

anyhow, what we have here are,
first of all, obviously, a couple graphs
and a bunch of code. here, at risk
of verbosity, is some line-by-line
commentary.

in the upper left is
part of the graph of
the linear equation
y=(x+1)/2…
namely the part whose x’s
(x co-ordinates) are between
-1 and 1.

and my students (like all
deserving pre-calculus graduates)
are familiar with *most* of the
notations… and *all* the ideas…
in this first line.

that funky *arrow*, though.
well, i can’t easily put it in here
(my wordpress skills are but weak)
but i’m talking about the one
looking otherwise like
.
and in the actual *zine*, it’s
a Bijection Arrow.
something like ” >—->>”.

as explained (or, OK, “explained”)
*below* the dotted line.
where *three* set-mapping “arrows”
are defined (one in each line;
the in each line
denotes “is equivalent-by-definition to”).

the Injective Arrow >—> denotes that
f:D—->R is “one-to-one” (as such
functions are generally known
in college maths [and also in
the pros for that matter; “injective”
and its relatives aren’t *rare*,
but their plain-language versions
still get used oftener]).

“one-to-one”, defined informally,
means “different x’s always get
different y’s”. coding this up
(“formally”), with D for the “domain”
and R for the “range” (though i
prefer “target” in this context
when i’m actually present to
*explain* myself) means that
when d_1 and d_2 are in D,
and d_1 \not= d_2
(“different x’s”), one has
f(d_1) \not= f(d_2)
(“different y’s”).

likewise the Surjective Arrow —>>
denotes what is ordinarily called
an “onto” function:
every range element
(object in R)
“gets hit by” some domain element.

and of course the Bijective Arrow >—>>
denotes “one-to-one *and* onto”.

there’s quite a bit of symbolism
here that’s *not* familiar to
typical college freshmen.
the Arrows themselves.
“for all”
“there exists”
logical “and”
and the seldom-seen-even-by-me
“such that” symbol that, again,
i’m unable to reproduce here
in type.

still, i hope i’m making a point
worth making by writing out
these “definitions without words”.

anyhow… worth doing or not…
it’s out of the way and i can return
to the main line of exposition:
the mapping from (-1, 1) to (0,1)
in the top line of my photo here
is a “bijection”, meaning that it’s
a “one-to-one and onto” function.

for *finite* sets A and B,
a bijection f: A —-> B
exists if and only if
# A = # B…
another unfamiliar notation
i suppose but readily understood…
A and B have
*the same number of elements*.
we extend this concept to *infinite*
sets… when A and B are *any*
sets admitting a bijection
f:A—>B, we again write
#A = #B
but now say that
A and B have the same cardinality
(rather than “number”;
in the general case, careful
users will pronounce #A
as “the cardinality of A”).

i’m *almost* done with the top line.
i think. but there’s one more notation
left to explain (or “explain”).

the “f:D—>R” convention i’ve been
using throughout this discussion
is in woefully scant use in textbooks.
but it *is* standard and (as i guess)
often pretty easily made out even
by beginners when introduced;
one has been *working* with
“functions” having “domains”
and “ranges”, so fixing the notation
in this way should seem pretty natural.

but replacing the “variable function”
symbol “f” by *the actual name
of the function* being defined?
this is *very rare* even in the pros.
alas.

the rest is left as an exercise.

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