Very good answer, Kim,
Just a few comments. and then the sequel.
>> Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
>> 3, ...}?
>
>
> No idea what square-root(2) means. When I said I was innumerate I
> wasn't kidding! I could of course look
> it up or ask my mathematics teacher friends but I just know your
> explanation will make theirs seem trite.
Well thanks.
The square root of 2 is a number x, such that x*x (x times x, x
multiplied by itself) gives 2.
For example, the square root of 4 is 2, because 2*2 is 4. The square
root of 9 is 3, because 3*3 is 9. Her by "square root" I mean the
positive square root, because we will see (more later that soon) that
numbers can have negative square root, but please forget this.
At this stage, with this definition, you can guess that the square
root of 2 cannot be a natural number. 1*1 = 1, and 2*2 = 4, and it
would be astonishing that x could be bigger than 2. So if there is
number x such that x*x is 2, we can guess that such a x cannot be a
natural number, that is an element of {0, 1, 2, 3 ...}, and the answer
of exercise 4 is "no". The square root of two will reappear
recurrently, but more in examples, than in the sequence of notions
which are strictly needed for UDA-7.
>> A set is entirely defined by its elements. Put in another way, we
>> will
>> say that two sets are equal if they have the same elements.
>> Exercise 6. Let S be the set {0, 1, 45} and let M be the set
>> described
>> by {45, 0, 1}. Is it true or false that S is equal to M?
>
>
> True - unless integer position within a given sequence in a set
> plays a role. I will guess that it does not.
You are right.
>> Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
>> true or false that S is equal to M?
>
>
> False - the commas separate each natural number
You are right. Also note that there is only one element in the set {6,
6, 6}. It is just a redundant description of the set {6}.
> Very excited about doing this. If you can make it all as
> approachable as this I am over the moon!
I will try, and it is very kind to play such a candid role. I
appreciate that you have the ability to say "I don't know
<something>". It is very helpful for me to remain approachable, and
eventually it will help everybody.
So let us continue.
=============== 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 = ?
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/
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