Re: The seven step-Mathematical preliminaries 2
Marty,
On 07 Jun 2009, at 02:03, Brent Meeker wrote:
m.a. wrote:
*Okay, so is it true to say that things written in EXTENSION are
never
in formula style but are translated into formulas when we put them
into INTENSION form? You can see that my difficulty with math
arises from an inability to master even the simplest definitions.
marty a.*
It's not that technical. I could define the set of books on my
shelf by
giving a list of titles: The Comprehensible Cosmos, Set Theory and
It's Philosophy, Overshoot, Quintessence. That would be a
definition by extension. Or I could point to them in succession and
say, That and that and that and that. which would be a definition by
ostension. Or I could just say, The books on my shelf. which is a
definition by intension. An intensional definition is a descriptive
phrase with an implicit variable, which in logic you might write as:
The
set of things x such that x is a book and x is on my shelf.
This is a good point. A set is just a collection of objects seen as a
whole.
A definition in extension of a set is just a listing, finite or
infinite, of its elements.
Like in A = {1, 3, 5}, or B = {2, 4, 6, 8, 10, ...}.
A definition in intension of a set consists in giving the typical
defining property of the elements of the set.
Like in C= the set of odd numbers which are smaller than 6. Or D =
the set of even numbers.
In this case you see that A is the same set as C? And B is the same
set as D.
Now in mathematics we often use abbreviation. So, for example, instead
of saying: the set of even numbers, we will write
{x such-that x is even}.
OK?
Bruno
Suppose,
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
Thank you, Brent, This is quite clear. Hopefully I can apply it as clearly to Bruno's examples.marty a. - Original Message - From: Brent Meeker meeke...@dslextreme.com To: everything-list@googlegroups.com Sent: Saturday, June 06, 2009 8:03 PM Subject: Re: The seven step-Mathematical preliminaries 2 m.a. wrote: *Okay, so is it true to say that things written in EXTENSION are never in formula style but are translated into formulas when we put them into INTENSION form? You can see that my difficulty with math arises from an inability to master even the simplest definitions. marty a.* It's not that technical. I could define the set of books on my shelf by giving a list of titles: The Comprehensible Cosmos, Set Theory and It's Philosophy, Overshoot, Quintessence. That would be a definition by extension. Or I could point to them in succession and say, That and that and that and that. which would be a definition by ostension. Or I could just say, The books on my shelf. which is a definition by intension. An intensional definition is a descriptive phrase with an implicit variable, which in logic you might write as: The set of things x such that x is a book and x is on my shelf. Brent --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Bruno,
Yes, this seems very clear and will be helpful to refer back to if
necessary. m.a.
- Original Message -
From: Bruno Marchal marc...@ulb.ac.be
To: everything-list@googlegroups.com
Sent: Sunday, June 07, 2009 4:33 AM
Subject: Re: The seven step-Mathematical preliminaries 2
Marty,
On 07 Jun 2009, at 02:03, Brent Meeker wrote:
m.a. wrote:
*Okay, so is it true to say that things written in EXTENSION are
never
in formula style but are translated into formulas when we put them
into INTENSION form? You can see that my difficulty with math
arises from an inability to master even the simplest definitions.
marty a.*
It's not that technical. I could define the set of books on my
shelf by
giving a list of titles: The Comprehensible Cosmos, Set Theory and
It's Philosophy, Overshoot, Quintessence. That would be a
definition by extension. Or I could point to them in succession and
say, That and that and that and that. which would be a definition by
ostension. Or I could just say, The books on my shelf. which is a
definition by intension. An intensional definition is a descriptive
phrase with an implicit variable, which in logic you might write as:
The
set of things x such that x is a book and x is on my shelf.
This is a good point. A set is just a collection of objects seen as a
whole.
A definition in extension of a set is just a listing, finite or
infinite, of its elements.
Like in A = {1, 3, 5}, or B = {2, 4, 6, 8, 10, ...}.
A definition in intension of a set consists in giving the typical
defining property of the elements of the set.
Like in C= the set of odd numbers which are smaller than 6. Or D =
the set of even numbers.
In this case you see that A is the same set as C? And B is the same
set as D.
Now in mathematics we often use abbreviation. So, for example, instead
of saying: the set of even numbers, we will write
{x such-that x is even}.
OK?
Bruno
Suppose,
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
*Bruno et. al., Good news! I have discovered that the math symbols copy faithfully here in my Thunderbird email.* *Henceforth, I will open all list letters here. Please refresh my memory for the following symbols:* * 1. The ***?** *is called_and means__ 2. The***?** *is called___*_*and means__ 3. The ***? is called__and means ** -* 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 ? ? A = ? ? B = A ? ? = B ? ? = N ? ? = B ? ? = ? ? B = ? ? ? = ? ? ? = * --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Bravo Thunderbird!
On 07 Jun 2009, at 18:39, m.a. wrote:
Bruno et. al.,
Good news! I have discovered that the math
symbols copy faithfully here in my Thunderbird email. Henceforth, I
will open all list letters here. Please refresh my memory for the
following symbols:
1. The ∅ is called___THE EMPTY SET_and means__THE SET
WITH NO ELEMENTS
The empty set described in extension: { }
The empty set described in intension. Well, let me think. The set of
french which are bigger than 42 km tall.
A cynical definition would be: the set of honest politicians.
A mathematical one: the set of x such that x is different from x.
It is just the set which has no elements. It is empty.
2. The∪ is calledUNION__and means: A ∪ B__= {x
such-that x belongs to A or x belongs to B};
A u B is the set obtained by doing the union of A and B.
3. The ∩ is called_INTERSECTIONand means__A ∩ B__=
{x such-that x belongs to A andr x belongs to B}; A u B is the
set obtained by doing the intersection of A and B. It is the set of
elements which are in both A and B._
Examples:
{1, 2, 3} ∩ {2, 4, 3} = {2, 3}
{1, 2, 3} u {2, 4, 3} = {1, 2, 3, 4}
{1, 2, 3} ∩ {4, 5, 6} = ∅
{1, 2, 3} u {4, 5, 6} = {1, 2, 3, 4, 5, 6}
OK?
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
∅ ∪ A =
∅ ∪ B =
A ∪ ∅ =
B ∪ ∅ =
N ∩ ∅ =
B ∩ ∅ =
∅ ∩ B =
∅ ∩ ∅ =
∅ ∪ ∅ =
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
Marty, Kim,
I realize that, now, the message I have just sent does not have the
right symbols. Apparently my computer does not understand the
Thunderbird!
From now on I will use capital words for the mathematical symbols.
And I will write mathematical expression in bold.
For examples:
{1, 2, 3} INTERSECTION {2, 4, 3} = {2, 3}
{1, 2, 3} UNION {2, 4, 3} = {1, 2, 3, 4}
{1, 2, 3} INTERSECTION {4, 5, 6} = EMPTY
{1, 2, 3} UNION {4, 5, 6} = {1, 2, 3, 4, 5, 6}
All right? Mathematics will get a FORTRAN look but this is not
important, OK? It is just the look. I will do a summary of what we
have seen so far.
With those notions you should be able to invent exercises by yourself.
Invent simple sets and compute their union, and intersection.
Remenber that the goal consists in building a mathematical shortcut
toward a thorugh understanding of step seven. In particular the goal
will be to get an idea of a computation is, and what is the difference
between a mathemarical computation and a mathematical description of a
computation. It helps for the step 8 too.
Marty, have a nice holiday,
Kim, ah ah ... we have two weeks to digest what has been said so far
(which is not enormous), OK?
Bruno
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
Bruno, Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below? - 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 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- 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 -~--~~~~--~~--~--~--- --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Marty, Bruno, Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below? We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday 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 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- 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 -~--~~~~--~~--~--~--- 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 -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries 2
If it helps, here's a screenshot of how the symbols are supposed to look: http://img34.imageshack.us/img34/3345/picture2uzk.png From: marc...@ulb.ac.be To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries 2 Date: Sat, 6 Jun 2009 22:36:01 +0200 Marty, Bruno, Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below? We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday Bruno - Original Message -From: Bruno MarchalTo: everything-l...@googlegroups.comsent: Wednesday, June 03, 2009 1:15 PMSubject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A =∅ ∪ B =A ∪ ∅ =B ∪ ∅ =N ∩ ∅ =B ∩ ∅ =∅ ∩ B =∅ ∩ ∅ =∅ ∪ ∅ = ---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 -~--~~~~--~~--~--~--- 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 -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
(I'll be here till Tuesday.) Evidently, the symbol you are using for such that is being shown on my screen as a small rectangle. In the copy below, I see two rectangles before the A=, two before the B=, two after the A, two after the B. The UNION symbol (inverted U) shows up but is followed by a rectangle in the next two examples and preceded by a rectangle in the last three. In checking a table of logic notaion, I find that the relation such that is designated by a reversed capital E. Is this the symbol you are using? m.a. - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Saturday, June 06, 2009 4:36 PM Subject: Re: The seven step-Mathematical preliminaries 2 We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday 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 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- 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 -~--~~~~--~~--~--~--- 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 -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Bruno,
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}.
Can you see the cause of my confusion? 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.
- 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 10x 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
Re: The seven step-Mathematical preliminaries 2
On 06 Jun 2009, at 23:54, m.a. wrote: (I'll be here till Tuesday.) Evidently, the symbol you are using for such that is being shown on my screen as a small rectangle. In the copy below, I see two rectangles before the A=, two before the B=, two after the A, two after the B. The UNION symbol (inverted U) shows up but is followed by a rectangle in the next two examples and preceded by a rectangle in the last three. In checking a table of logic notaion, I find that the relation such that is designated by a reversed capital E. Is this the symbol you are using? m.a. Yes, we have a problem. There should be no rectangles at all. We have to switch on english abbreviations. This explains the difficulty you did have with the union ... You could look on the archive, from here, http://www.mail-archive.com/everything-list@googlegroups.com/msg16531.html the symbols are correct on my computer, but we will think on easier mail symbols. Tell me if you see different symbols in the archive. Best, Bruno - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Saturday, June 06, 2009 4:36 PM Subject: Re: The seven step-Mathematical preliminaries 2 We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday 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 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- 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 -~--~~~~--~~--~--~--- 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 -~--~~~~--~~--~--~---
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
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 10x 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
Re: The seven step-Mathematical preliminaries 2
On this date, you made the following correction: You cannot write D = 4*x
..., But you wrote D= 4*x in the exercise just above it. I don't get
the distinction between your use of the equation and mine.
- Original Message -
From: Bruno Marchal
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 10x is a multiple of 4}.D=4*x where x = 0 (but also 1,2,3...10)
You cannot write D = 4*x ..., given that D is a set, and 4*x is a (unknown)
number (a multiple of four when x is a natural number).
Read carefully the problem. I gave the set in intension, and the exercise
consisted in writing the set in extension. Let us translate in english the
definition of the set D = {x ⎮ x 10x is a multiple of 4}: it means that
D is the set of numbers, x, such that x is little than 10, and x is a multiple
of four. So D = {0, 4, 8}.
SEE BELOW
Example: the set of multiple of 4 is {0, 4, 8, 12, 16, 20, 24, 28, 32, 36,
...}, all have the shape 4*x, with x = to 0, 1, 2, 3, ...
The set of multiple of 5 is {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55,
...}
Etc.
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.
In the example above (that is {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}) we
were taking the INTERSECTION of the two sets.
But after that, may be too quickly (and I should have made a title perhaps) I
was introducing the UNION of the two sets.
If you read carefully the definition in intension, you should see that the
intersection of A and B is defined with an and. The definition of union is
defined with a or. Do you see that? It is just above in the quote.
I hope that your computer can distinguish A ∩ B (A intersection B) and A ∪ B
(A union B).
In the union of two sets, you put all the elements of the two sets together.
In the intersection of two sets, you take only those elements which belongs to
the two sets.
It seems you have not seen the difference between intersection and union.
This has indeed been the case. My usual math disabilities have been
exacerbated by the confusion of symbols due to E-mail limitations. The
profusion of little rectangles replacing the UNION symbol make the formulae
difficult to follow.
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Re: The seven step-Mathematical preliminaries 2
m.a. wrote:
*Bruno,*
* 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}.*
No, that's not the intensional definition. This We can easily define A
in intension: A = the set of numbers which are even and more little
than 100. is the intensional definition.
Brent
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Re: The seven step-Mathematical preliminaries 2
Bruno, When I tried to copy the symbols from the URL cited below, I found that my email server was not able to reproproduce the intersection or the union symbol. See below: From: Bruno Marchal To: everything-list@googlegroups.com ∅ ∪ A = I see two rectangles and A ∅ ∪ B = I see two rectangles and B A ∪ ∅ = I see A and two rectangles B ∪ ∅ = I see B and two rectangles N ∩ ∅ = I see N Inverted U and a rectangle B ∩ ∅ = I see B Inverted U and a rectangle ∅ ∩ B = I see a rectangle an inverted U and B ∅ ∩ ∅ = I see a rectangle an inverted U and a rectangle ∅ ∪ ∅ = I see three rectangles - Original Message - From: Bruno Marchal You could look on the archive, from here, http://www.mail-archive.com/everything-list@googlegroups.com/msg16531.html --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Okay, so is it true to say that things written in EXTENSION are never in
formula style but are translated into formulas when we put them into INTENSION
form? You can see that my difficulty with math arises from an inability to
master even the simplest definitions.marty a.
- Original Message -
From: Bruno Marchal
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.
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
m.a. wrote: *Okay, so is it true to say that things written in EXTENSION are never in formula style but are translated into formulas when we put them into INTENSION form? You can see that my difficulty with math arises from an inability to master even the simplest definitions. marty a.* It's not that technical. I could define the set of books on my shelf by giving a list of titles: The Comprehensible Cosmos, Set Theory and It's Philosophy, Overshoot, Quintessence. That would be a definition by extension. Or I could point to them in succession and say, That and that and that and that. which would be a definition by ostension. Or I could just say, The books on my shelf. which is a definition by intension. An intensional definition is a descriptive phrase with an implicit variable, which in logic you might write as: The set of things x such that x is a book and x is on my shelf. Brent --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Bruno, Thanks for the encouragement. I intend to follow your instructions and it's a relief to know that some of my answers were correct. However, I will be away for two weeks and unable to work on the lessons. I'll try to make up for it when I return. Best, marty a. - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Friday, June 05, 2009 10:03 AM Subject: Re: The seven step-Mathematical preliminaries 2 Hi Marty, On 05 Jun 2009, at 00:30, m.a. wrote: Bruno, I don't have dyslexia Good news. but my keyboard doesn't contain either the UNION symbol or the INTERSECTION symbol Nor do mine! (unless I want to go into an INSERT pull down menu every time I use those symbols). Like I have to do too. I don't need you to switch to English symbols, but I would like to see the English equivalents of the symbols you use (so that I can use them). I gave them. I would also like a reference table defining each term in both your symbols and their English equivalents which I could look back to when I get confused. I suggest you do this by yourself. It is a good exercise and it will help you not only in the understanding, but in the memorizing. Then you submit it to the list, and I can verify the understanding. Please include examples. Up to now, I did it for any notions introduced. Just ask me one or two or (name your number) examples more in case you have a doubt. If I send too much posts, and if there are too long, people will dismiss them. try to ask explicit question, like you did, actually. I tend to be somewhat careless when dealing with very fine distinctions This means that a lot of work is awaiting for you. It is normal. Everyone can understand what I explain, but some have more work to do. and may type the wrong symbol while intending to type the correct one. That is unimportant. I am used to do typo errors too. One of my favorite book on self-reference (the one by Smorynski) contains an average of two or three typo error per page. Of course, once a typo error is found, it is better to correct it. Also, I must admit that the lessons are going too fast for me and are moving ahead before I've mastered the previous material. We have all the time, and up to now I did not proceed without having the answer of all exercises. You make no faults in the first set of seven exercise, and that is why I have quickly proceed to the second round. For that one, you make just one error, + the dismiss of a paragraph on UNION. To slow me down it is enough to tell me things like I don't understand what you mean by this or that and you quote the unclear passage. If you can't do an exercise, just wait for some other (Kim?) to propose a solution. Or try to guess one and submit, or just ask. I will not proceed to new matters before I am sure you grasp all what has been already presented. What is possible is that you understand, but fail ti memorize. This will lead to problems later. So you have to make your own summary and be sure you can easily revise the definition. If I'm requesting too much simplification, please let me know because I'm quite well adjusted to my math disabilities and won't take offence at all. Thanks, marty a. I think that there is no problem at all. I am just waiting for explicit question from the second round. You can ask any question, and slow me down as much as you want so that we proceed at your own rhythm. Don't ask me to slow down in any abstract way. You are the one who have to slow me down by pointing on what you don't understand in a post. take it easy, and take all your time. Don't try to understand the more advanced replies I give to people who have a bigger baggage. You did show me that you have understood the notion of set, and the notion of intersection of sets. Have you a problem with the notion of union of sets? If that is the case, just quote the passage of my post that you don't understand, or the example that I gave, and I will explain. Try to keep those post in some well ranged place so as to re-access them easily. I ask this to Kim too, and any one interested: just let me know what you don't understand, so that I can explain, give other examples, etc. Take it easy, you seem quite good, you suffer just of a problem of familiarity with notations. You read the post too quickly, I suspect also. Are you OK? I can understand you could be afraid of the amount of work, but given that we have all the time, there is no exams, nor deadline, I am not sure there is any problem
Re: The seven step-Mathematical preliminaries 2
On Thu Jun 4 1:15 , Bruno Marchal sent:
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.
OK - I find this quite mind-blowing; probably because I now understand it for
the first
time in my life. So how did it get this rather ridiculous name of square
root? What's it
called in French?
(snip)
=== 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 an 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 smaller than 100. Mathematicians will
condense
this by the following:
A = {x such that x is even and smaller than 100} = {x ⎮ x is even x
special character, abbreviating such that, and I hope it goes through the
mail.
Just an upright line? It comes through as that. I can't seem to get this symbol
happening so I will
use such that
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 objects x, (or number x if we are in a context
where we talk
about numbers) 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 and 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 and x is a multiple
of 4}?
D = 4*x where x = 0 but also { 1, 2, 3, 4, 8 }
I now realise I am doomed for the next set of exercises because I cannot get to
the special
symbols required (yet). As I am adding Internet Phone to my system, I am
currently using an
ancient Mac without the correct symbol pallette while somebody spends a few
days to flip a single
switch...as soon as I can get back to my regular machine I will complete the
rest.
In the meantime I am enjoying the N+1 disagreement - how refreshing it is to
see that classical
mathematics remains somewhat controversial!
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Re: The seven step-Mathematical preliminaries 2
Hi Marty,
On 04 Jun 2009, at 01:11, m.a. wrote:
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.
No problem.
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}
I guess you meant C = {x such that x is odd and x 101}. means
bigger than, and means little than. OK.
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 10x is a multiple of
4}.D=4*x where x = 0 (but also 1,2,3...10)
You cannot write D = 4*x ..., given that D is a set, and 4*x is a
(unknown) number (a multiple of four when x is a natural number).
Read carefully the problem. I gave the set in intension, and the
exercise consisted in writing the set in extension. Let us translate
in english the definition of the set D = {x ⎮ x 10x is a
multiple of 4}: it means that D is the set of numbers, x, such that x
is little than 10, and x is a multiple of four. So D = {0, 4, 8}.
Indeed 0 is little than 10, and 0 is a multiple of four (because 0 =
4*0), and
4 is little than 10, and 4 is a multiple of four (because 4 = 4*1)
8 is little than 10, and 8 is a multiple of 4 (because 8 = 4*2)
The next mutiple of 4 is 12. It cannot be in the set because 12 is
bigger than 10.
The numbers 1, 2, 3, 5, 7, 9 cannot be in D, because they are not
multiple of 4. You cannot write 1 = 4 * (some natural numbers), nor
can you write 3 or 5, or 7 or 9 = 4 * x with x a natural number.
Example: the set of multiple of 4 is {0, 4, 8, 12, 16, 20, 24, 28, 32,
36, ...}, all have the shape 4*x, with x = to 0, 1, 2, 3, ...
The set of multiple of 5 is {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50,
55, ...}
Etc.
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.
In the example above (that is {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5,
6}) we were taking the INTERSECTION of the two sets.
But after that, may be too quickly (and I should have made a title
perhaps) I was introducing the UNION of the two sets.
If you read carefully the definition in intension, you should see that
the intersection of A and B is defined with an and. The definition
of union is defined with a or. Do you see that? It is just above in
the quote.
I hope that your computer can distinguish A ∩ B (A intersection B)
and A ∪ B (A union B).
In the union of two sets, you put all the elements of the two sets
together. In the intersection of two sets, you take only those
elements which belongs to the two sets.
It seems you have not seen the difference between intersection and
union. I guess you try to go to much quickly, or that the font of
your computer are too little, or that you have eyesight problems, or
that you have some dyslexia.
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 x10}= {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}
All that would be correct if you were searching the intersection, but
∪ is the UNION symbol. (and ∩ is the INTERSECTION symbol).
also you wrote the ⎮ as inter, instead of such that.
N ∩ A = {0,1,2,3...} inter {x inter x10}= {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}
All that is correct. Careful you were still using inter in place of
such that. Your last line should be
A ∩ B = {x such that x 10} inter {x such that 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
Both are correct.
Not bad Marty! Just read carefully. I
Re: The seven step-Mathematical preliminaries 2
On Thu, Jun 4, 2009 at 7:28 AM, kimjo...@ozemail.com.au
kimjo...@ozemail.com.au wrote:
On Thu Jun 4 1:15 , Bruno Marchal sent:
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.
OK - I find this quite mind-blowing; probably because I now understand it for
the first
time in my life. So how did it get this rather ridiculous name of square
root? What's it
called in French?
I don't know what it is called in French, but I can answer the first
part. I remember the day I first figured out where the term came
from.
When you have a number multiplied by itself, the result is called a
square. 3*3 = 9, so 9 is a square. Imagine arranging a set of peas,
if you can arrange them in a square (the four cornered kind) with the
same number of rows as columns, then that number is a square. Some
examples of squares are: 4, 9, 16, 25, 36, 49, 64, 81, see the
pattern? And the roots of those squares are 2, 3, 4, 5, 6, 7, 8,
and 9. The square root is equal to the number of items in a row, or
column when you arrange them in a square.
This is a completely extraneous fact, but one I consider to be very
interesting: Multiply any 4 consecutive positive whole numbers and the
result will always be 1 less than a square number. For example,
5*6*7*8 = 1680, which is 1 less than 1681, which is 41*41. Isn't that
neat?
Jason
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Re: The seven step-Mathematical preliminaries 2
Hi Kim,
On 04 Jun 2009, at 14:28, kimjo...@ozemail.com.au wrote:
OK - I find this quite mind-blowing; probably because I now
understand it for the first
time in my life. So how did it get this rather ridiculous name of
square root? What's it
called in French?
Racine carrée. Literally square root.
It comes from the fact that in elementary geometry the surface or area
of a square which sides have length x, is given by x*x, also written
x^2, which is then called the square of x. Taking the square root
of a number, consists in doing the inverse of taking the square of a
number. It consists in finding the length of a square knowing its area.
Mathematician and especially logician *can* use arbitrary vocabulary.
It is the essence of the axiomatic method in pure mathematics that
what is conveying does not depend on the term which are used. Hilbert
said once that he could have use the term glass of bear instead of
line in his work in geometry.
A = {x such that x is even and smaller than 100} = {x ⎮ x is even
x
special character, abbreviating such that, and I hope it goes
through the mail.
Just an upright line? It comes through as that. I can't seem to get
this symbol happening so I will
use such that
Yes, such that is abbreviated by an upright line. Sometimes also by
a half circle followed by a little line, but I don't find it on my
palette!
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 objects x, (or number x if we are in
a context where we talk
about numbers) 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 and x 101}
Correct.
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 and x is
a multiple of 4}?
D = 4*x where x = 0 but also { 1, 2, 3, 4, 8 }
Hmm...
Marty made a similar error. D is a set. May be you wanted to say:
D = {4*x where x = 0 but also { 1, 2, 3, 4, 8 }}. But this does not
make much sense. Even if I try to imagine favorably some meaning, I
would say that it would mean that D is the set of numbers having the
shape 4*x (that is capable of being written as equal to 4*x for some
x), and such that x belongs to {0, 1, 2, 3, 4, 8}.
A proper way to describe that set would be
D = {y such that y = 4x and x belongs-to {0, 1, 2, 3, 4, 8}}.
But that would makes D = {0, 4, 8, 12, 32}.
The set D = {x ⎮ x 10 and x is a multiple of 4} is just, in
english, the set of natural numbers which are little than 10 and which
are a multiple of 4. The only numbers which are little than 10, and
multiple of 4 are the numbers 0, 4, and 8. D = {0, 4, 8}.
I now realise I am doomed for the next set of exercises because I
cannot get to the special
symbols required (yet). As I am adding Internet Phone to my system,
I am currently using an
ancient Mac without the correct symbol pallette while somebody
spends a few days to flip a single
switch...as soon as I can get back to my regular machine I will
complete the rest.
Take it easy. No problem.
In the meantime I am enjoying the N+1 disagreement - how refreshing
it is to see that classical
mathematics remains somewhat controversial!
The term is a bit too strong. It is a bit like if I told you that I
am Napoleon, and you conclude that the question of the death of
Napoleon is still controversial. I exaggerate a little bit to make my
point, but I know only two ultrafinitists *in math*, and I have never
understood what they mean by number, nor did I ever met someone
understanding them.
What makes just a little bit more sense (and I guess that's what
Torgny really is) is being ultrafinitist *in physics*, and being
physicalist. You postulate there is a physical universe, made of a
finite number of particles, occupying a finite volume in space-time,
etc. Everything is finite, including the everything.
Then by using the unintelligible identity thesis (and thus
reintroducing the mind-body problem), you can prevent the comp white
rabbits inflation. Like all form of materialism, this leads to
eliminating the person soon or later (by the unsolvability of the mind-
body problem by finite means). Ultrafinitist physicalism eliminates
also mathematics and all immaterial notions, including all universal
machines. Brrr...
The real question is do *you* think that there is a biggest natural
number? Just tell me at once, because if you really believe that
there is a biggest natural number, I have no more clues at all how you
could believe in any of computer science nor UDA.
Remember that Thorgny pretends also to be a zombie. It has already
Re: The seven step-Mathematical preliminaries 2
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 10x 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
Re: The seven step-Mathematical preliminaries 2
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@googlegroups.com
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 10x 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 x10}= {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
