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 02 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, 03 ...}, 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 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 and x is a multiple of 4}?
D = 4*x where x = 0 but also { 1, 2, 3, 4, 08 }
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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Received on Thu Jun 04 2009 - 20:28:20 PDT