Hi, Bruno,
let me skip the technical part and jump on the following text.
*F u n c t i o n* as I believe is - for you - the y = f(x) *form*. For me:
the *activity -* shown when plotting on a coordinate system the f(x) values
of the Y-s to the values on the x-axle resulting in a relation (curve). And
here is my problem: who does the plotting? (Do not say: YOU are, or Iam,
that would add to the function concept the homunculus to make it from a
written format into a F U N C T I O N ).
John M
On Wed, Jul 29, 2009 at 11:59 AM, Bruno Marchal <marchal.domain.name.hidden> wrote:
> SOLUTIONS
>
>
> OK. I give the solution of the exercises of the last session, on the
> cartesian product of sets.
>
> I recall the definition of the product A X B.
>
> A X B = {(x,y) such that x belongs to A and y belongs to B}
>
> I gave A = {0, 1}, and B = {a, b}.
>
>
> In this case, A X B = {(0,a), (0, b), (1, a), (1, b)}
>
> The cartesian drawing is, for AXB :
>
>
> a (0, a) (1, a)
>
> b (0, b) (1, b)
>
> 00 1
>
>
> Exercise: do the cartesian drawing for BXA.
>
> Solution:
>
> 01 (a, 1) (b, 1)
>
> 0 (a, 0) (b, 0)
>
> a b
>
> You see that B X A = {(a,0), (a,1), (b,0), (b, 1)}
>
> You should see that, not only A X B is different from B X A, but AXB and
> BXA have an empty intersection. They have no elements in common at all. But
> they do have the same cardinal 2x2 = 4.
>
> 1)
> Compute
> {a, b, c} X {d, e} =
> I show you a method (to minimize inattention errors):
>
> I wrote first {(a, _), (b, _), (c, _), (a, _), (b, _), (c, _)} two times
> because I have seen that {d, e} has two elements.
> Then I add the second elements of the couples, which comes from {d, e}:
>
> {(a, d), (b, d), (c, d), (a, e), (b, e), (c, e)}
>
> OK?
>
>
> {d, e} X {a, b, c} = {(d, a), (d, b), (d, c), (e, a), (e, b), (e, c)}
>
> {a, b} X {a, b} = {(a, a), (a, b), (b, a), (b, b)}
>
> {a, b} X { } = { }.
>
> OK?
>
> 2)
> Convince yourself that the cardinal of AXB is the product of the
> cardinal of A and the cardinal of B.
> A and B are finite sets here. Hint: meditate on their cartesian drawing.
>
> Question? This should be obvious. No?
>
>
> 3) Draw a piece of NXN. (with, as usual, N = {0, 1, 2, 3, ...}):
>
> . . . . . . . .
> . . . . . . . .
> . . . . . . . .
> 05 (0,5) (1,5) (2,5) (3,5) (4,5) (5,5) ...
> 04 (0,4) (1,4) (2,4) (3,4) (4,4) (5,4) ...
> 03 (0,3) (1,3) (2,3) (3,3) (4,3) (5,3) ...
> 02 (0,2) (1,2) (2,2) (3,2) (4,2) (5,2) ...
> 1 (0,1) (1,1) (2,1) (3,1) (4,1) (5,1) ...
> 0 (0,0) (1,0) (2,0) (3,0) (4,0) (5,0) ...
>
> 0 1 2 3 4 5 ...
>
>
> OK?
>
>
> N is infinite, so N X N is infinite too.
>
>
> Look at the diagonal: (0,0) (1,1) (2,2) (3,3) (4,4) (5,5) ...
>
> definition: *the diagonal of AXA,* a product of a set with itself, is the
> set of couples (x,y) with x = y.
>
> All right? No question? Such diagonal will have a quite important role in
> the sequel.
>
> Next: I will say one or two words on the notion of relation, and then we
> will define the most important notion ever discovered by the humans: the
> notion of function. Then, the definition of the exponentiation of sets, A^B,
> is very simple: it is the set of functions from B to A.
> What is important will be to grasp the notion of function. Indeed, we will
> soon be interested in the notion of computable functions, which are mainly
> what computers, that is universal machine, compute. But even in physics, the
> notion of function is present everywhere. That notion capture the notion of
> dependency between (measurable) quantities. To say that the temperature of a
> body depends on the pressure on that body, is very well described by saying
> that the temperature of a body is a function of the pressure.
> Most phenomena are described by relation, through equations, and most
> solution of those equation are functions. Functions are everywhere, somehow.
>
> I have some hesitation, though. Functions can be described as particular
> case of relations, and relations can be described as special case of
> functions. This happens many times in math, and can lead to bad pedagogical
> decisions, so I have to make a few thinking, before leading you to
> unnecessary complications.
>
> Please ask questions if *any*thing is unclear. I suggest the "beginners" in
> math take some time to invent exercises, and to solve them. Invent simple
> little sets, and compute their union, intersection, cartesian product,
> powerset.
> You can compose exercises: for example: compute the cartesian product of
> the powerset of {0, 1} with the set {a}. It is not particularly funny, but
> it is like music. If you want to be able to play some music instrument,
> sometimes you have to "faire ses gammes",we say in french; you know, playing
> repetitively annoying musical patterns, if only to teach your lips or
> fingers to do the right movement without thinking. Math needs also such a
> kind of practice, especially in the beginning.
> Of course, as Kim said, passive understanding of music (listening) does not
> need such exercises. Passive understanding of math needs, alas, many
> "simple" exercises. Active understanding of math, needs difficult exercises
> up to open problems, but this is not the goal here.
>
> Bruno
>
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
>
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Received on Thu Jul 30 2009 - 11:53:41 PDT