Partial Order - Reconciling Definitions by Garling and Goldrei ....

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SUMMARY

The discussion centers on the differing definitions of partial orders as presented by D. J. H. Garling in "A Course in Mathematical Analysis: Volume I" and Derek Goldrei in "Classic Set Theory for Guided Independent Study." Garling's definition omits reflexivity, while Goldrei includes it, leading to questions about the implications of this difference. Stephen Tashi clarifies that Garling's condition (ii), which emphasizes anti-symmetry, implies reflexivity when specific cases are considered. This highlights a non-standard approach by Garling that may still hold value in certain contexts.

PREREQUISITES
  • Understanding of mathematical analysis concepts, particularly relations and orders.
  • Familiarity with anti-symmetry and reflexivity in the context of set theory.
  • Knowledge of D. J. H. Garling's and Derek Goldrei's works in mathematical literature.
  • Basic comprehension of equivalence relations and their properties.
NEXT STEPS
  • Study the implications of reflexivity in partial orders and its relationship with anti-symmetry.
  • Examine the definitions of partial equivalence relations (PERs) and their applications.
  • Explore the nuances of mathematical definitions in foundational texts like Garling's and Goldrei's.
  • Investigate the role of common language in mathematical definitions and proofs.
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Mathematics students, educators, and researchers interested in set theory, particularly those studying relations and orders in mathematical analysis.

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I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... and I am also referencing concepts in Derek Goldrei's book, "Classic Set Theory for Guided Independent Study" ...

I am currently focused on Garling's Section 1.3 Relations and Partial Orders ... ...

Garling defines a partial order as follows:

View attachment 6138
... ... BUT Goldrei's definition is (apparently) slightly different ... as follows:

View attachment 6139

Can anyone explain why Goldrei includes reflexivity but Garling doesn't ... ... ?

Is it because reflexivity can be derived somehow from Garling's condition (ii) ... which appears to simply be anti-symmetry ... ?

Can someone please clarify this issue ...

Help will be appreciated ...

Peter
 
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The definition without reflexivity seems nonstandard. I've never seen it. Maybe it's useful in some areas, similarly to how partial equivalence relations (PERs) are useful.
 
Evgeny.Makarov said:
The definition without reflexivity seems nonstandard. I've never seen it. Maybe it's useful in some areas, similarly to how partial equivalence relations (PERs) are useful.
Thanks Evgeny ... it really makes me wonder why Garling did it ...

It is not as if D. J. H. Garling is not experienced or eminent ... he is an Emeritus Reader in mathematical analysis at the University of Cambridge and he has 50 years of experience teaching undergraduates ... ...

Thanks again for our information ... and your thought on why Garling defined a partial order in this way ... I thought it must be that reflexivity flowed from condition (ii) ...

Seems to me a strange decision (by a really experienced teacher!) to put a non-standard definition in a book designed and written for undergraduates ...

Peter
 
Peter said:
I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... and I am also referencing concepts in Derek Goldrei's book, "Classic Set Theory for Guided Independent Study" ...

I am currently focused on Garling's Section 1.3 Relations and Partial Orders ... ...

Garling defines a partial order as follows:... ... BUT Goldrei's definition is (apparently) slightly different ... as follows:
Can anyone explain why Goldrei includes reflexivity but Garling doesn't ... ... ?

Is it because reflexivity can be derived somehow from Garling's condition (ii) ... which appears to simply be anti-symmetry ... ?

Can someone please clarify this issue ...

Help will be appreciated ...

Peter
I thought that MHB readers would be interested in the following post on the Physics Forums ... ...

In answer to my question:

" ... ... Is it because reflexivity can be derived somehow from Garling's condition (ii) ... which appears to simply be anti-symmetry ... ? ... ... "


[h=3]Stephen Tashi[/h] writes:" ... ... Yes. Condition (ii) says "if and only if". So if we take the case a=b=x , condition (ii) implies a≤b, which is equivalent to "x≤x".

(An interesting technical question is whether this a consequence of the definition of "=" for some particular equivalence relation, or whether it is a consequence of the "common language" definition of the relation "=", which , in common mathematical speech implies "You can substitute one of a pair of "equal" symbols for another in any symbolic expression in a proof.")

... ... ... ... "Do MHB readers agree?Peter
 
Last edited:
Of course reflexivity follows from "if" of (ii). Sorry I did not notice it earlier.
 

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