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 <[email protected]> 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) > > 0 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 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/ > > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
