# Re: The seven step-Mathematical preliminaries 2

```
>            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}".```
```

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
> 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
> 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
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> http://iridia.ulb.ac.be/~marchal/
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http://iridia.ulb.ac.be/~marchal/

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