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jackson6612

Member

Registered: 2006-07-19

Posts: 5

order properties

HI

Please your answer simple. Thank you

Order Properties ( of real numbers ):

The properties satisfied by the relation < ( less than ) in the field R of real numbers. The basic properties are:

(1): Trichotomy law: if r and s are real numbers then one and only one of the statements r < s, r = s and s < r holds.

(2): Transitive law: if r, s, and t are real numbers r < s and s < t and r < t.

(3): If r < s then r+u < s+u for any real number u.

(4): If r < s and u is real number, then ru < su if u > 0.

(5): Completeness property: any nonempty set of numbers that is bounded above has a least upper bound.

The first four properties above are summarized by saying that R is an ordered field. There are other ordered fields. For instance, the rational numbers satisfy (1) to (4) ( reading 'rational' for 'real' each time ), but R is the only ordered field which also has the completeness property (5), i.e. is a complete field. Every nonempty set of real numbers that is bounded below ( has a lower bound ) must have a greatest lower bound.

Question(s):

1: [3/4, 5/6, 7/8, 9/7] is subset of rational number and has 9/7 as least upper bound?

2: Please explain above given definition of 'order properties' in some detail by giving numerical example. Thank you

Sincerely,
vijay

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: order properties

(2): Transitive law: if r, s, and t are real numbers r < s and s < t and r < t.

(3): If r < s then r+u < s+u for any real number u.

(4): If r < s and u is real number, then ru < su if u > 0.

Once you properly define > and <,  the above become theorems, not laws.

[3/4, 5/6, 7/8, 9/7] is subset of rational number and has 9/7 as least upper bound?

Yes, if a set S contains a max, then that max is the least upper bound (or supremum, or sup) of the set.  9/7 is the max of that set, so it is also the least upper bound.

2: Please explain above given definition of 'order properties' in some detail by giving numerical example. Thank you

I'm not sure way you mean.  The properties are fairly intuitive.

5 < 6 < 7, so 5 < 7
5 < 6, 5 + 3 < 6 + 3
6 < 6.000001 u=0.0000001, then 6*0.0000001 < 6.000001*0.0000001 (try it on a calculator)

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"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..."

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: order properties

The last property is called the continuousity of Real Numbers which is unique, some mathematicians have a great trick to prove it, so I suggest you to get a book involving cantor set. Real Analysis perhaps.

The upper bound should be in the set of real numbers, instead of in the sub set.

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X'(y-Xβ)=0

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: order properties

That's funny George, I've always heard of it as Completeness.  Oh well, po-ta-toe, po-tot-oe I guess.

My Real Analysis book starts out calling it an axiom, then changes it to a theorem at the end.  Since I've never read it all the way through, I can't tell you how they got there.

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"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..."

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: order properties

Uh, the completeness, I guess, refer to all of the 5. Cproperty is proposed by Dedkind and absorbed by Cantor.

If it is really called completeness, better.

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X'(y-Xβ)=0

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: order properties

No, I don't believe so.  Completeness is just the inclusion of the sup and inf of any set inside the number system you are working in.  Wikipedia has a lot on it, and I don't quite have to time to read it now.  Maybe tomorrow.

Of course, it's just a name.

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"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..."