Re: Sets question (maybe)
- From: "cbrown@xxxxxxxxxxxxxxxxx" <cbrown@xxxxxxxxxxxxxxxxx>
- Date: Tue, 23 Oct 2007 10:44:55 -0700
On Oct 23, 7:22 am, Name And Address Supplied
<name_and_address_suppl...@xxxxxxxxxxx> wrote:
Dear all,
I wonder if you can help with the following problem. I'm not a
mathematician, so please bear with me.
Informally speaking: I have a set of, say, 5 entities, and I wish to
give each a unique index j. In particular, I want to pick one entity
-- it doesn't matter which -- and I want to designate that j = 1. Then
I want to pick another, and index that one j = 2. And so on until I
have five individuals indexed 1 to 5.
Note that this is the same thing as assigning some element of the 5
entities to each of 1,2,3,4,5.
And that's the same thing as defining a function S with S(1) = (the
element of from the 5 entities you picked first), S(2) = (the element
of J you picked second), etc.
So in a way, you are talking about a /function/: a special sort of
function with domain {1,2,3,4,5} and range the 5 entities.
Now I want to order this list,
so that it runs from 1 through to 5.
Isn't it /already/ ordered by observing that, for example, 1 < 2, 2 <
3, etc.?
Q1: What do I call this list?
In mathematics, it's usually referred to as a finite sequence (or just
a list).
If J is a sequence, whose elements are taken from the set S, we will
usually write J = (j_1, j_2, j_3, ..., j_n), where n is the "number of
entities"; and where "j_1" is how we write "j with subscript 1" in
sci.math. Each j_i is some member of S.
Thinking of J as a function, J(1) = j_1, J(2) = j_2, etc.
Note that in mathematics, a list can also refer to an /infinite/
sequence (j_1, j_2, ..., j_n, ...).
I think it is not a set in the strict
sense, as a set is not ordered, and also a set with repeated elements
is equal to the same set with the repeated elements removed.
That is correct. The standard way to approach this issues is as
follows:
{x, y} = {y, x} as sets: two sets are th same if, and only if, they
have the same members.
So we instead /define/ (x, y) to be the /set/ {x, {x, y}}. This is
"the ordered pair (x,y)". A little thought (well, plus the fact that
it is forbidden for a set to have itself as a member) should convince
you that (x, y) = (y, x) if, and only if, x = y.
So we can then think of the finite /sequence/ J = (j_1, j_2, ..., j_5)
as the /set/ of ordered pairs S = {(1, j_1), (2, j_2), ..., (5, j_5)}.
This is exactly the same (in set theory) as defining J to be a /
function/ which takes the naturals {1,2,3,4,5} (often just written as
"the set 5") to the set J. So we can also write "J : 5 -> S", and talk
about J(3) being the third element of the sequence J, etc.
But I
want to describe the object J = {1, 2, 3, 4, 5}...
Careful! Is J the set of /entities/ (whatever they might be)? The
above sort of confuses J with the set of /indexes/...
, and say things like
"j E J" (j is a member of J).
We might loosely say "j is a member of the sequence J = (j_1, j_2,
j_3, ..., j_n)" to mean the more rigorously correct "there exists i in
(1, 2, .., n} such that j = j_i", or equivalently (given the above
definitions) "... such that (i, j) in J" or "... such that j = J(i)".
I also want to say that the jth element
of J is j...
Careful - you're again confusing your /indexes/ with your /entities/!
It is also possible that j is /not/ the jth element of J.
-- this of course would not be true if J wasn't ordered /
had repeated elements. So, what is J? A set, a list, a group, what?
It's a sequence or list, which is a special sort of function (it has a
range which is from 1 to some natural), which is a special sort of
set.
Q2: I don't know that there are actually 5 entities, but I do know
there is a finite number of them. I want to index them as above, and I
want to say in a mathematically compact way that the indices are the
natural numbers running from 1 to the maximum index value (which is
equal to the number of entities I want to index). In particular, I
don't want to have to assign a symbol to denote the total number of
entities.
If S is the set of entities, one often writes |S| (which is spoken as
"the cardinality of J" or "card of J") to refer to the number of
(distinct!) elements in S. Then you can talk about the finite sequence
J = {j_1, j_2, ..., j_|S|}; or equivalently consider J as a function
J : |S| -> S, with j_i = J(i).
Q3: If J were a set, and K = {1,2} were some other set, I could define
a Cartesian product JxK = {{{1,1},{1,2}},{{2,1},{2,2}},{{3,1},{3,2}},
{{4,1},{4,2}},{{5,1},{5,2}}}.
That's /not/ the Cartesian product J x K. The Cartesian product is
the /set/ of all /ordered pairs/ of the form (j,k), j in J, k in K. It
has |J|*|K| = 10 elements. What you have written above has elements of
the form {{j, 1}, {j, 2}} for j in J. It has only |J| = 5 elements;
each such element having |K| = 2 elements.
But if J and K are not sets, but are the
type of objects I've described above, can I still make this operation?
Without a bit more context, it's not clear exactly /what/ it is you're
trying to invoke with this operation. My guess is that, given natural
numbers a, b, you want to be able to talk about the ordered pair (j_a,
k_b), where j_a is the "a'th" element in the sequence J, and k_b is
the "b'th" element of the sequence K.
You could express that as a function T which maps each possible pair
(a,b) to the corresponding pair (j_a, k_b). In that case (recalling
that J and K are functions in the set theory sense) you can just
define the function T(a,b) = (J(a), K(b)).
As a /set/, T would then have elements of the form ((a,b),
(J(a),K(b)); an ordered pair of ordered pairs.
Is that what you're trying to formalize?
Or is there an analogous operation, and if so, what is it called?
Q4: Is the "curly bracket" notation okay for what I am doing, i.e. "{"
and "}"?
No. You want to use "(" and ")" to indicate that you are talking about
a finite sequence. Sometimes people alternatively use "<" and ">" if
there are lots of other parentheses kicking around already; for
example J = <j_1, j_2, j_3, ..., j_n>.
Cheers - Chas
.
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