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:
1 (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, ...}):
. . . . . . . .
. . . . . . . .
. . . . . . . .
5 (0,5) (1,5) (2,5) (3,5) (4,5) (5,5) ...
4 (0,4) (1,4) (2,4) (3,4) (4,4) (5,4) ...
3 (0,3) (1,3) (2,3) (3,3) (4,3) (5,3) ...
2 (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 01 02 03 04 05 ...
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 Wed Jul 29 2009 - 17:59:26 PDT