MHB Antisymmetry & Partial Orderings - H&J Ch.2 Section 5 | Peter's Help

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Antisymmetry and reflexivity are distinct concepts in set theory, as antisymmetry requires that if both \(aRb\) and \(bRa\) hold, then \(a\) must equal \(b\), while reflexivity only requires that \(aRa\) for all \(a\). The discussion highlights the importance of rigor in understanding these definitions, emphasizing that informal reasoning can lead to confusion. Examples of antisymmetric relations include non-strict and strict inequalities, the divisibility relation on integers, and specific cases on real numbers. The conversation encourages deeper analysis and precise formulation of these concepts to clarify their differences. Understanding these distinctions is crucial for grasping the foundations of partial orderings.
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I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ...

I am currently focused on Chapter 2: Relations, Functions and Orderings; and, in particular on Section 5: Orderings

I need some help with H&J's depiction of antisymmetric relations and partial orderings ...The introduction to H&J's section on antisymmetric relations and partial orderings reads as follows:View attachment 7599In the above text from H&J we read the following:

" ... ... A binary relation $$R$$ in $$A$$ is antisymmetric if for all $$a, b \in A$$, $$aRb$$ and $$bRa$$ imply $$a = b$$ ... "Now since $$a = b$$ in the above instance $$aRb$$ and $$bRa$$ can be expressed as $$aRa$$ (or $$bRb$$ ...) ... so isn't antisymmetry essentially reflexivity ...

Can someone explain to me exactly what the essential difference between antisymmetry and reflexivity ... and, indeed why we don't define partial orderings simply as relations that are reflexive and transitive ... Help will be appreciated ...

Peter
 
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Peter said:
In the above text from H&J we read the following:

" ... ... A binary relation $$R$$ in $$A$$ is antisymmetric if for all $$a, b \in A$$, $$aRb$$ and $$bRa$$ imply $$a = b$$ ... "Now since $$a = b$$ in the above instance $$aRb$$ and $$bRa$$ can be expressed as $$aRa$$ (or $$bRb$$ ...) ... so isn't antisymmetry essentially reflexivity
I think you should think about this more rigorously. The definition of antisymmetry does not say $a=b$ (and for which $a$ and $b$?). If you think there is an equivalent, simpler statement of antisymmetry, try to write it precisely and then prove that it is equivalent to the original definition. Considering examples of relations that are and are not antisymmetric also helps.
 
Evgeny.Makarov said:
I think you should think about this more rigorously. The definition of antisymmetry does not say $a=b$ (and for which $a$ and $b$?). If you think there is an equivalent, simpler statement of antisymmetry, try to write it precisely and then prove that it is equivalent to the original definition. Considering examples of relations that are and are not antisymmetric also helps.
Hi Evgeny,

Yes I was being somewhat informal ...

I should have said ... since aRb and bRa imply a = b ... then we can essentially write aRa or bRb ... that is the condition aRb and bRa gives us reflexivity ... hmm ... yes ... see your point ... what exactly am i saying ...

Have to rethink my question ...

By the way ... can you think of a good example of an antisymmetric relation ...?

Peter
 
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Peter said:
can you think of a good example of an antisymmetric relation ...?
Both non-strict and strict inequalities on numbers are antisymmetric, as are the divisibility relation on integers and the empty relation. Another one is $$\begin{cases}x<y,&x\ge0,y\ge0\\x\le y,&\text{otherwise}\end{cases}$$ on real numbers.
 
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