Re: Objects and Relations

On Jan 31, 11:23 am, "Neo" <neo55..._at_hotmail.com> wrote:
> > > I still can't understand the logic/rationale explaining the transition
> > > from sets with unordered elements to sets with ordered elements,
> > > either from above or Chapter 7 of Schuam's Outline.
>
> > Hint: there *are no* sets with ordered elements. There are sets
> > of unordered elements, and there are *other* sets of unordered
> > elements that contain information that specifies an order for another
> > set.
>
>
> The above two statements are contradictory. First you say set elements
> have no order. Then you say a second set whose elements are also
> unordered imply the order of the elements of the first set. But this
> is contradictory as we already established in the beginning that the
> elements of the first or any set are unordered.

There is no contradiction.

It is possible that you are thinking of sets as variables, such that when
we create the second order relation we somehow change the nature of the first relation. This is not the mathematical perspective as I understand
it. Sets are immutable values. Creating a second relation doesn't change
the nature of the first relation.

There is no "transition."

Here's a set of integers:

F = {1, 2}

Here's a binary relation on (integers x integers) that specifies an order
on F.

O1 = { (1,2) }

Did the creation of O1 somehow change the nature of F? No, it remains exactly {1, 2} as it was before, with no order on the elements.

However, what if we consider the ***pair*** (F, O1). Please note that the
pair (F, O1) is not a set; it is an ordered pair. That pair can be interpreted
as an "ordered set" (even though it's not a set) with two elements, in order,
first 1, then 2.

Alternate syntax for (F, O1)

  [1, 2]

Now consider this set:

O2 = { (2, 1) }

Note that both O2 and O1 exist now, they both specify an order on S, although neither
of them *modifies* S in any way. We only get an "ordered set" when we consider
(F, O1) or (F, O2) as an ordered pair.

You also have to master the syntax:

Inside {} are the elements of a set, which have no meaningful order Inside () are the attributes of an ordered tuple (or "pair" if it's a 2-tuple)
Inside [] are the elements of a list, which have meaningful order.

{} sets are homogeneous.
() are heterogeneous.
[] are homogeneous.

This means you can freely reorder the things inside {} and consider it the same value. However you CANNOT reorder the things inside () nor [].

Until you master the syntax you don't stand a chance of understanding the important distinctions to master this issue.

> Let me try to apply your "system":
>
> I am assuming { {1, apple}, {2, orange}, {3, banana} } would imply the
> order apple, orange and then banana.

Nope! Already wrong. Unless you fix the syntax, your example is broken. Note that fixing the syntax changes the meaning of what you wrote.

If you want something that's an order for {apple, orange, banana} then that would be something like

  {(1, apple), (2, orange), (3, banana)}

The thing you wrote is a set of sets. However what is required is a set of ordered pairs.

> Since you said set elements are
> ordered, what order does the following imply: {{1, apple}, {orange,
> 2}, {3, banana}}

You can freely reorder the elements of {2, orange} from your example. However that doesn't say anything about order because the *reason* you can reorder those elements is because you wrote it as a set of sets. But an order is a set of ordered pairs. You *can't* reorder the elements of (2, orange). (Without making it into a different value, the same way you get a different value when you rewrite the characters of "12" into "21".

Good luck. Please be sure to read carefully, and especially note the difference between {}, (), and [].

Marshall

> And if { {1, 100}, {2, 34}, {3, 200} } implies the order 100, 34 and
> then 200. The would the following should imply the same: { {34, 2},
> {3, 200}, {1, 100} }
Received on Wed Jan 31 2007 - 23:18:32 CET