> I've encountered some difficulty with the examples below. > You say that "in extension" describes exhaustion or quasi- > exhaustion. And you give the example: "B = {3, 6, 9, 12, ... 99}". > Then you define "in intension" with exactly the same type > of set: "Example: Let A be the set {2, 4, 6, 8, 10, ... 100}".

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I give A in extension there, but just to define it in intension after. It is always the same set there. But I show its definition in extension, to show the definition in intension after. You have to read the to sentences. > Can you see the cause of my confusion? It is always the same set. I give it in extension, and then in intension. > Incidentally, may I suggest you use "smaller than" rather than > "more little than". Your English is generally too good to include > that kind of error. marty a. Well sure. Sometimes the correct expression just slip out from my mind. "smaller than " is much better! Thanks for helping, Bruno > > > > > ----- Original Message ----- > From: Bruno Marchal > To: everything-list@googlegroups.com > Sent: Wednesday, June 03, 2009 1:15 PM > Subject: Re: The seven step-Mathematical preliminaries 2 > > > =============== Intension and extension ==================== > > > > In the case of finite and "little" set we have seen that we can > define them by exhaustion. This means we can give an explicit > complete description of all element of the set. > Example. A = {0, 1, 2, 77, 98, 5} > > When the set is still finite and too big, or if we are lazy, we can > sometimes define the set by quasi exhaustion. This means we describe > enough elements of the set in a manner which, by requiring some good > will and some imagination, we can estimate having define the set. > > Example. B = {3, 6, 9, 12, ... 99}. We can understand in this case > that we meant the set of multiple of the number three, below 100. > > A fortiori, when a set in not finite, that is, when the set is > infinite, we have to use either quasi-exhaustion, or we have to use > some sentence or phrase or proposition describing the elements of > the set. > > Definition. > I will say that a set is defined IN EXTENSIO, or simply, in > extension, when it is defined in exhaustion or quasi-exhaustion. > I will say that a set is defined IN INTENSIO, or simply in > intension, with a "s", when it is defined by a sentence explaining > the typical attribute of the elements. > > Example: Let A be the set {2, 4, 6, 8, 10, ... 100}. We can easily > define A in intension: A = the set of numbers which are even and > more little than 100. mathematician will condense this by the > following: > > A = {x such that x is even and little than 100} = {x ⎮ x is even & > x < 100}. "⎮" is a special character, abbreviating "such that", and > I hope it goes through the mail. If not I will use "such that", or > s.t., or things like that. > The expression {x ⎮ x is even} is literally read as: the set of > object x, (or number x if we are in a context where we talk about > number) such that x is even. > > Exercise 1: Could you define in intension the following infinite set > C = {101, 103, 105, ...} > C = ? > > Exercise 2: I will say that a natural number is a multiple of 4 if > it can be written as 4*y, for some y. For example 0 is a multiple of > 4, (0 = 4*0), but also 28, 400, 404, ... Could you define in > extension the following set D = {x ⎮ x < 10 & x is a multiple of > 4}. > > A last notational, but important symbol. Sets have elements. For > example the set A = {1, 2, 3} has three elements 1, 2 and 3. For > saying that 3 is an element of A in an a short way, we usually write > 3 ∈ A. this is read as "3 belongs to A", or "3 is in A". Now 4 > does not belong to A. To write this in a short way, we will write 4 > ∉ A, or we will write ¬ (4 ∈ A) or sometimes just NOT(4 ∈ A). > It is read: 4 does not belong to A, or: it is not the case that 4 > belongs to A. > > Having those notions and notations at our disposition we can speed > up on the notion of union and intersection. > > The intersection of the sets A and B is the (new) set of those > elements which belongs to both A and B. Put in another way: > The intersection of the sets A with the set B is the set of those > elements which belongs to A and which belongs to B. > This new set, obtained from A and B is written A ∩ B, or A inter. B > (in case the special character doesn't go through). > With our notations we can write or define the intersection A ∩ B > directly > > A ∩ B = {x ⎮ x ∈ A and x ∈ B}. > > Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6} > > Similarly, we can directly define the union of two sets A and B, > written A ∪ B in the following way: > > A ∪ B = {x ⎮ x ∈ A or x ∈ B}. Here we use the usual > logical "or". p or q is suppose to be true if p is true or q is true > (or both are true). It is not the exclusive "or". > > Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}. > > Exercice 3. > Let N = {0, 1, 2, 3, ...} > Let A = {x ⎮ x < 10} > Let B = {x ⎮ x is even} > Describe in extension (that is: exhaustion or quasi-exhaustion) the > following sets: > > N ∪ A = > N ∪ B = > A ∪ B = > B ∪ A = > N ∩ A = > B ∩ A = > N ∩ B = > A ∩ B = > > Exercice 4 > > Is it true that A ∩ B = B ∩ A, whatever A and B are? > Is it true that A ∪ B = B ∪ A, whatever A and B are? > > Now, I could give you exercise so that you would be lead to > discoveries, but I prefer to be as simple and approachable as > possible, and my goal is not even to give you the taste for doing > research, so I will do the discovery by myself here and now. Indeed > a natural question occurs. What will happen if we try to find the > intersection of two sets which have no elements in common? For > example, what is the intersection of A = {x ⎮ x is even} with B = > {x ⎮ x is odd} ? At first sight we could say that there is no > intersection, given that A and B have no elements in common. But a > set is just a bit more than its elements. And if there is no > elements in the intersection, it means simply that the set A ∩ B > has no elements. So we are very inspired if we let that bizarre set > to exist, so we give it a name, and call it the empty set, and we > can describe it easily in exhaustion by { }, although many describe > it as ∅. So, if A and B have no elements in common, A ∩ B is > still well defined and is equal to ∅. having a new toy, we can play > with it: > > Exercise 5, with A and B the same as in exercise 3. > > ∅ ∪ A = > ∅ ∪ B = > A ∪ ∅ = > B ∪ ∅ = > N ∩ ∅ = > B ∩ ∅ = > ∅ ∩ B = > ∅ ∩ ∅ = > ∅ ∪ ∅ = > > > ----------------------- > SUBSET > We will say that A is a subset of B (A and B being sets) if, > whatever object x represents, each time x belongs to A, it belongs > to B. Put in another way it means that IF x belongs to A, THEN x > belongs to B. It means that all the elements of A are also elements > of B. We can write, with > > x ∈ A -> x ∈ B. > > And this we abbreviate as A ⊆ B, and we read it: A is included in B. > > Example: > 1) Let us look if the set A = {1, 2} is included in the set B = {1, > 2, 3}. Here A has two elements. To see if A is included in B, we > have to look at each element in the set A, and we have to see if > they belongs to B. Now A has two elements, 1, and 2, so we have two > tasks to accomplish, or two questions to answer: > does 1 belongs also to B. The answer is yes. > does 2 belongs also to B. The answer is yes. > We have thus verify that all elements of A are also elements of B, > and thus we can conclude that A is indeed included in B. > > 2) Let us look if the set A = {1} is included in B = {1, 2, 3}. > Now, A has only one element. So we are lucky, we have only one task > to accomplish! Is 1 an element of B? The answer is yes. Thus we have > {1} is included in {1, 2, 3}. > > 3) Let us look if the set A= { }, the empty set ∅, is included in > B = {1, 2, 3}. Now A has no element. So we are even more lucky, we > have no task to accomplish at all. The condition is trivially > satisfied. So the empty set is included in {1, 2, 3}. And this shows > that the empty set is included in any set. In particular we have > that ∅ ⊆ ∅. > Note that all set is a subset of itself. Trivially, all elements of > A is an element of A. > > Exercise 6 > We will say that a set A is a subset of a set B, if A is included in > B. > Could you give all the subsets of the set {1, 2}. > Could you give all the subsets of the set {1} > Could you give all the subsets of the set { }. > > The post is long enough, so I spare you the seventh exercise. Also I > have to go, I hope there are not to many typo errors and spelling > mistakes, and well, I pray for the special symbols going trough. It > is possible that they go through for most mailing systems, but not > all. Let me know. > > Bon courage, > > Bruno > > > > > > > > > > > > > > > http://iridia.ulb.ac.be/~marchal/ > > > > > > > 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 -~----------~----~----~----~------~----~------~--~---