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: [email protected]
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 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}. 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, 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}. 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, 7 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 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/
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