Partial Order .... Garling, pages 9-10, Volume I ,,,

In summary, Garling's definition of a partial order in his book "A Course in Mathematical Analysis: Volume I" requires transitivity and the condition that a and b are equal if and only if a is less than or equal to b and b is less than or equal to a. This definition differs from that of other authors, such as Cunningham, who define a partial order as also including reflexivity. However, it is possible to derive reflexivity from Garling's definition.
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I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...

I am focused on Chapter 1: The Axioms of Set Theory ... ...

I need some help to clarify an aspect of Garling's definition of a partial order ...Garling's definition of a partial order reads as follows:View attachment 9028
View attachment 9029Garling thus defines a partial order relation as possessing transitivity and anti-symmetry ... but a number of other authors define a partial order as having the property of reflexivity as well ...

... ... an example is Daniel W. Cunningham: Set Theory: A First Course ... who defines a partial order as follows:View attachment 9030Can someone explain why Garling does not include reflexivity in his definition of a partial order ...?

Is it perhaps that it is possible to derive reflexivity from transitivity and anti-symmetry ... but how do we do that ...?Help will be appreciated ...

Peter
 

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  • #2
Note that the definitions of antisymmetry in Garling and Cunningham differ. Garling does not even call it antisymmetry.
 
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Evgeny.Makarov said:
Note that the definitions of antisymmetry in Garling and Cunningham differ. Garling does not even call it antisymmetry.
Oh! Indeed, Evgeny ... think I see what you mean ...

After transitivity, Garling's second condition for a partial order is as follows:

(ii) \(\displaystyle a \leq b\) and \(\displaystyle b \leq a \ \ \) if and only if \(\displaystyle \ \ a = b\) ... so if \(\displaystyle a = b\) we have \(\displaystyle a \leq a\) ... that is, we have reflexivity ... I should have read Garling's condition more carefully ... Thanks for the help, Evgeny ...

Peter
 

FAQ: Partial Order .... Garling, pages 9-10, Volume I ,,,

1. What is a partial order?

A partial order is a mathematical concept that defines a relationship between elements of a set. It is a binary relation that is reflexive, antisymmetric, and transitive.

2. How is a partial order different from a total order?

A total order is a special case of a partial order where all elements in a set are related to each other. In a partial order, some elements may not be related to each other, making it a more general concept.

3. What is the importance of partial orders in mathematics?

Partial orders are important in many areas of mathematics, including set theory, graph theory, and order theory. They are also used in computer science and other fields to model relationships between objects.

4. How are partial orders represented?

Partial orders can be represented in various ways, such as Hasse diagrams, directed graphs, and matrices. These representations help visualize the relationships between elements in a set.

5. What are some real-world applications of partial orders?

Partial orders have many practical applications, such as in scheduling and ranking tasks, organizing data, and analyzing decision-making processes. They are also used in fields like economics, social sciences, and engineering.

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