Bruno,
I stopped half-way through because I'm not at all sure of my answers and would like to have them confirmed or corrected, if necessary, rather than go on giving wrong answers. marty a.
----- Original Message -----
From: Bruno Marchal
To: everything-list.domain.name.hidden
Sent: Wednesday, June 03, 2009 1:15 PM
Subject: Re: The seven step-Mathematical preliminaries 2
=============== Intension and extension ====================
Before defining "intersection, union and the notion of subset, I would like to come back on the ways we can define some specific sets.
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 = ? C={x such that x is odd & x <101}
Exercise 2: I will say that a natural number is a multiple of 04 if it can be written as 4*y, for some y. For example 00 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}. D=4*x where x = 0 (but also 1,2,3...10)
A last notational, but important symbol. Sets have elements. For example the set A = {1, 2, 3} has three elements 1, 02 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}. Question: In the example above, 5,6 were the intersection because they were the (only) two numbers BOTH groups had in common. But in this example, 07 is only in the second group yet it is included in the answer. Please explain.
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 = {0,1,2,3...} inter {x inter x<10}= {0,1,2,3...9}
N ∪ B = {0,1,2,3....} inter {x inter x is even}= {0,2,4,6...}
A ∪ B = {x inter x <10} inter {x inter x is even}= {0,2,4,6,8}
B ∪ A = {x inter x is even} inter {x inter x < 10}= {0,2,4,6,8}
N ∩ A = {0,1,2,3...} inter {x inter x<10}= {0,1,2,3...9}
B ∩ A = {x inter x is even} inter {x inter x < 10}= {0,2,4,6,8}
N ∩ B = {0,1,2,3....} inter {x inter x is even}= {0,2,4,6...}
A ∩ B = {x inter x <10} inter {x inter x is even}= {0,2,4,6,8}
Exercice 4
Is it true that A ∩ B = B ∩ A, whatever A and B are? yes
Is it true that A ∪ B = B ∪ A, whatever A and B are? yes
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 01 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/
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Received on Wed Jun 03 2009 - 19:11:49 PDT