## Archive for November, 2010

1.Let A and B denote sets. Use an “element” argument to prove the “Absorbing Law” .

**Proof:**

Suppose first that . It follows from the definition of intersection that then . Since was chosen arbitrarily in , we have shown that .

Now suppose instead that . By the definition of union, we can then say that also . These two statements together (and the definition of intersection) now give us . So ( was arbtrary… we are “generalizing from the generic particular” again) we’ve shown that .

The two inclusions together show that .

*Part of the point here is that “element” style proofs are direct but somewhat tedious. For “higher level” set equations, one begins with a few well-established “laws” (like this “absorbing” law) and uses them to write out a string of equations (rather than “chasing” an element from side to side as we have done here).*

2.Let the Universal Set (for this problem) be the Irrational Numbers (R – Q) and let and .

Compute the given sets.

*Should be a gift at our level of the game; one sees similar problems in Math for Poets classes: this is * very *basic set theory. Still, I anticipate lots of errors… like leaving off the set braces altogether among others. Did I mention that I’m avoiding actually* grading *these darn things?*

3.Let A and B denote sets Show algebraically that . Give a reason for each equation.

**Proof:**

(Set Difference Law)

(De Morgan’s Law)

(Double Complement Law)

(Commutative Law)

*Much more fun than “chasing” an element like in 1.*

4.Let X = {p, q} and compute the sets (the power set) and (the cartesian product of X with itself).

P(X) = {

{ }, {p}, {q}, {p,q}

}

.

*One typically does* not * see the cartesian product… which I call simply the “cross” product… in the “math for poets” (terminal-introductory) classes. This is too bad. Anyhow, this is basic-basic stuff again; with any luck, the class’ll’ve flattened it.*

5.Write out (as a set of ordered pairs) the “less than or equal relation” given by on the set .

R = {

(0,0), (0,1), (0,2),

(1,1), (1,2),

(2,2)

}

*With, ideally, every comma and parenthesis in place. One may of course write out R = {(0,0), (0,1), (0,2), (1,1), (1,2), (2,2)} but it’s harder that way.*

6.Consider the function given by (for all positive integersn). Name the domain & codomain forf. Isfone-to-one? (Prove your answer.) Isfonto? (Prove your answer.)

(*I meant to have had f(n) = n^2. Rats.*)

The domain is the set of positive integers.

The codomain is the set of integers.

The function *f* is one-to-one. **Proof.** Suppose f(a) = f(b) [for some positive integers a and b]. Since f is the *identity* function (Rats!), we have f(a) = a and f(b)=b;

it follows that a = b. But is the defining condition for “one-to-one”.

But *f* is *not* onto. To show this, we must “construct” an integer (element of the codomain) which isn’t in the “image” (or “range”) of *f*. So consider . If f(x) = -1 for some *x* in the domain, we would have x = -1 by the definition of *f*. But -1 is *not* a positive integer. So there is no such *x*.

7.Now consider given by (for all real numbersx). Repeat the questions of the previous problem (forgrather than forf).

*This would have been more interesting if I hadn’t messed up the previous problem… the point is that the answers depend, not only on the “formula” given, but also on the domain and codomain.*

The domain and codomain are of course and respectively (the Real Numbers and the *Nonegative* Reals).

The squaring function is *not* one-to-one over the Reals; for example (-2)^2 = 2^2 [but -2 \not= 2; note that both *are* Reals]. The squaring function *is* onto the non-negative reals. To see this, let . Then and ; done.

*For the record, the squaring function* is *one-to-one when the domain is “zee-plus” (the positive integers) and is * not * onto the set of integers for this domain. Rats.*

in my other blog (you may have to scroll a little).

“midterm” report.

actually we’re more like two-thirds

of the way through (my one-shot

Discrete Math class at Big State U;

try to keep up)… but the custom

on this campus is to refer to

all-exams-other-than-the-final

as “midterms” (which strikes me

weird since where i came up we

sometimes had bigger-fuss-than-usual

exams in the *middle of the term*

[and called ’em “midterm” exams]).

maybe it’s a semesters-versus-quarters

thing. (but i’ll bet you money that

this usage persists long after the

upcoming switchover to semesters…)

i’ll go ahead and remark since it’s

on my mind that i still haven’t been

compensated financially for any of

this work. we walk by faith not sight.

there’s, what else, this third-party

computer interface whereby one is

required to divulge information

about one’s (so-called) bank account;

supposedly certain events then

occur over the internet… “automatic

deposits” i’ve heard ’em called…

and one’s later transactions are

“covered”. well, i’ve tried to

co-operate. twice now; we’ll see

with what success. anyhow, by some

miracle my phone (don’t get me started)

eventually… days late… delivered a

“voicemail” message from somebody in

(the tellingly nay damningly named)

“human resources” office… somehow

the e-check was refused at my bank and so

(according to this guy, on the phone, later)

they’ve sent an actual 3-D hardcopy check.

to my official address where i go

only about once a week so i’ll see

about it on monday. it’ll be about

a third of the quarter’s pay and

much the most money i’ve earned

in over a year so i’m pretty jazzed.

meanwhile, there’s a pile of grading:

the second “midterm”. on writing proofs.

and i’ll have ’em marked up by tuesday

evening when class meets but for now

i’m gonna let ’em age a bit.

the first exam was about logic and went

quite well… this is a *very* well-prepared

group by my standards (even in the pros

[from ’92 to ’96] i sort of specialized

in the survey-for-nonmajors stuff; once

i was sent down to the minors, it was

mostly “remedial” stuff) and the course

is pretty well-designed for students

at their level of the game.

my best move was finding the online course notes

for the guy who co-ordinated the course a few

quarters back (and taught at least one section

of his own): there is, more or less of course,

*much more material* covered by the sections

of the text we’re given to go over this quarter

than one can find time in class to discuss,

so i’ve followed this much-more-experienced

instructor’s choices of topics-to-stress

(as revealed in his exam-prep problem sets)

pretty closely and, so far knock wood,

with pretty good results.

meanwhile, the whole thing is overall

much mathier than the only *other*

Discrete Math class i ever taught

(back in the pros at Churchy Small

Liberal Arts College [now styling

itself a “University” but i digress]).

that one was more about compter-head

stuff like algorithms and data structures.

they’re quite a bit alike in requiring

students to be very careful with

definitions and stuff like that

i suppose… and i haven’t looked

at my notes from that long-ago

stuff for quite a while so they

might be even more alike than i think…

but some readers might naively believe

there’s some actual area of study

called “discrete math” and i’ll just

go ahead and strongly hint here that

they think it in error.

you get enough stuff in one of these

monster textbooks to keep even the

brightest of a class of college kids

busy for *years*. and for much the same

reason that to get a decent phone,

you’ve gotta sign on for years at

a time (even if you have no very

realistic idea of being able to

*pay* for it for that long let

alone of getting any useful service

without working like a wageslave

for untold amounts of very frustrating

time and no certain reward whatever):

“more waste faster” rules the roost.

but like i said. don’t get me started.

anybody worthy of the name “former math prof”

could bang out a text that would cost a small

fraction of what ours does and serve their

class just as well. i’ve done something

very like this myself for one of those

survey-for-nonmajors courses. there’s

even some overlap between that long-ago

work and the course at hand (as i’ve already,

in some sense, remarked).

and anybody worthy of name

“fit to survive in the present

economic climate” could probably

find a way to make it *pay*.

but concerning that of which

we cannot speak, it’s best

to STFU.