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#1 2006-07-30 00:58:05

jackson6612
Member
Registered: 2006-07-19
Posts: 5

partial order

Hi

Please keep your answer simple. Thank you

Note on notation:

<= for "less than or equal" and >= for "greater than or equal" and != as a symbol for "not equal".


Partial Order:

A relation <= between the elements of a set S that satisfies the following three conditions:

1: Reflexive condition: a <= a for each a in S.

2: Antisymmetric condition: for a and b in S, a <= b and b <= a can both hold only if a = b.

3: Transitive condition: if a, b, and c are in S, then a <= b and b <= c together imply a <= c.

If b <= a, then also a >= b; and if a <= b but a != b then a < b. An example of a set with a partial order is the set of natural numbers with n <= m if and only if n divides m.

If every pair of elements a, b in the set is comparable ( i.e. either a <= b or b <= a ) then the partially ordered set ( poset ) is called totally ordered or chain. The set of natural numbers is not totally ordered since, for example, 3 and 5 are not comparable. An example of a totally ordered set is the set of real numbers with the relation <= being the ordinary 'less than or equal to' relation.



I don't understand reflexive condition. I know about reflexive relation though.

Reflexive relation: A relation R on a set A is reflexive if, for all a ( belonging to ) A, a R a. The relation 'identity', for example, is reflexive on the set of natural numbers as every member is identical with itself.


Questions:

1: Can you give me example of any other relation other than 'identity' which is reflexive?

2: What does this ''reflexive condition a <= a for each a in S'' means? Please give me some simple numerical example.

3: What does that mean ''an example of a set with a partial order is the set of natural numbers with n <= m if and only if n divides m.''? What will happen if n does not divide m?

4: What does that mean ''the set of natural numbers is not totally ordered since, for example, 3 and 5 are not comparable''? I think 3 and 5 are comparable and we can write 3<5.


Sincerely,
vijay

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#2 2006-07-30 02:37:46

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: partial order

1: Can you give me example of any other relation other than 'identity' which is reflexive?

Every equivalence relationship must be reflexive.

This is sounding like abstract algebra, is that the course you're taking?  If so, then you should know group isomorphism at this point:

G ≈ H means that there exists a 1-1 and onto map from G to H such that θ(ab) = θ(a)θ(b).

≈ is an equivalence relationship, and more specifically:

G ≈ G

Thus, ≈ is reflexive.

It you don't know what I'm talking about, then just say so, there are many simpiler examples.  I just thought this one might have something to do with what you're doing.

2: What does this ''reflexive condition a <= a for each a in S'' means? Please give me some simple numerical example.

5 <= 5.  Honestly, that's all it really means.

3: What does that mean ''an example of a set with a partial order is the set of natural numbers with n <= m if and only if n divides m.''? What will happen if n does not divide m?

Well, the naturals are a totally ordered set, so I'm not quite sure either.

4: What does that mean ''the set of natural numbers is not totally ordered since, for example, 3 and 5 are not comparable''? I think 3 and 5 are comparable and we can write 3<5.

That is just wrong.  It got me doubting myself so much that I had to check Wikipedia:

The natural numbers comprise the smallest totally ordered set with no upper bound.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2006-07-30 13:17:13

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: partial order

I think natural numbers are cardinal, and comparable for sure.

A, B, C is sometimes uncomparable.


X'(y-Xβ)=0

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#4 2006-07-30 15:16:10

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: partial order

A, B, C is sometimes uncomparable.

What are A, B, and C?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#5 2006-07-30 15:24:34

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: partial order

If A, B, and C are real numbers, isn't it required that they be comparable by trichotomy and transitivity? Or am I wrong? Does trichotomy only say that there exists a definite relation between two real numbers, but says that it is not necessarily able to be found? If so, can you give an example?

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#6 2006-07-30 15:52:29

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: partial order

Social scientists as well as economists tend to categorize things into several groups according to their measurability. Here are measurability standard:
1. Can be categorized or can be classed . That means you can categorize some of them as As and some as Bs and so on. for example, cats and dogs. A and B is enough to simblize them, 1 and 2 are ok but brings on ambiguity.
2 Can be compared which means the categories are grouped according the extent of one property. for example, a person's height, experts grading.
3 Categories can be added freely and the added result is the simple sum of the added two.
4 Have an absolute zero value meaning null or nothing.

A variable satisfying 1 and 2 is called a cardinal variable or grade variable.
A variable satisfying all of the 5 is ideal, but rarely available in social science .

Natural numbers satisfy all 5 and definitely has absolute grading. 5 is larger and extremer than 3 for sure.


X'(y-Xβ)=0

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#7 2006-07-30 16:01:28

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: partial order

Zhylliolom, I believe trichotomy says that all reals must be comparable.  But I know just enough on the subject of reals to know that I don't know much on the subject of reals.

George, so you mean to define A, B, and C abstractly, correct?  They can be anything?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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