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 everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---