Proof of the properties of an ordered feild

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Homework Help Overview

The discussion revolves around proving a property of ordered fields, specifically that if \( b < a \), then \( -a < -b \). The original poster is questioning whether the definition of an ordered field is necessary for their proof, which utilizes basic axioms of fields.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to prove the property using basic axioms and questions the necessity of the ordered field definition. Other participants point out that properties like commutativity are part of the field definition and that the ability to add the same element to both sides of an inequality is also a characteristic of ordered fields.

Discussion Status

The discussion is exploring the relationship between the axioms used in the proof and the definition of an ordered field. Participants are clarifying the foundational aspects of the definitions and properties involved, indicating a productive exploration of the topic without reaching a consensus.

Contextual Notes

There is an ongoing examination of the definitions and axioms related to fields and ordered fields, with some participants emphasizing the importance of these definitions in the context of the proof.

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Homework Statement


If F is an ordered field the the following property holds for any elements a and b of F.
If b<a, the -a<-b.

My task is to prove this property. My question is whether I need to use the definition of an ordered field. I used the basic axioms but I didn't use the definition of an ordered field.


Homework Equations


The basic axioms such... communitivity, addative inverse...
The definition of an ordered field.


The Attempt at a Solution



Assume b<a. Then add -a-b to each side, which gives us b-a-b<a-a-b. by using communitivity on the left we can rewrite it as b-b-a<a-a-b. By the use of the addative inverse we can simplify it to -a<-b.(QED)

So does this work without the definition of an ordered field?
 
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Those properties, commutativity etc are part of the definition of a field, aren't they?
 
And the fact that you can add the same thing to both sides of an inequality is part of the definition of an ordered field.
 
The "definition of an ordered field" is that it is a set of objects, together with two operations and an order relation, that satisfy those axioms!
 

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