What are the properties of partial order relations according to J&W's book?

In summary, Peter is trying to prove that every irreflexive, transitive binary relation is asymmetric. However, he is having difficulty proving this and needs help from others.
  • #1
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I am reading the book: "Discovering Modern Set Theory. I The Basics" (AMS) by Winfried Just and Martin Weese.

I am currently focused on Chapter 2 Partial Order Relations ...

I need some help with Exercise 1(a) ... indeed, I have been unable to make a meaningful start on the exercise ... :(

The relevant section from J&W is as follows:View attachment 7542As mentioned above ... I have been unable to make a meaningful start on Exercise 1(a) ... can someone please help me with this exercise ... ... Help will be much appreciated ... ...

Peter===================================================================================It may be helpful for MHB members to have access to J&W's definitions of the properties of relations ... so I am providing the relevant text ... as follows:
View attachment 7543Hope that helps ...

Peter
 
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Peter said:
I am reading the book: "Discovering Modern Set Theory. I The Basics" (AMS) by Winfried Just and Martin Weese.

I am currently focused on Chapter 2 Partial Order Relations ...

I need some help with Exercise 1(a) ... indeed, I have been unable to make a meaningful start on the exercise ... :(

The relevant section from J&W is as follows:As mentioned above ... I have been unable to make a meaningful start on Exercise 1(a) ... can someone please help me with this exercise ... ... Help will be much appreciated ... ...

Peter===================================================================================It may be helpful for MHB members to have access to J&W's definitions of the properties of relations ... so I am providing the relevant text ... as follows:
Hope that helps ...

Peter

I have been reflecting on Example 1(a) of Chapter 2 in Just and Weese (see above)

I have an idea regarding a proof ... but I am most unsure that it is valid ...Now we have to show that every irreflexive, transitive binary relation R is asymmetric ...So ... we have to show that for \(\displaystyle R\) ... \(\displaystyle \langle a,b \rangle \in R \Longrightarrow \langle b, a \rangle \notin R\) ...So ... my attempt at a proof is as follows:Assume \(\displaystyle \langle a,b \rangle \in R\) ...NOW ... ALSO ASSUME that \(\displaystyle \langle b, a \rangle \in R\) ... and look for a contradiction ...... so ... now we have ...\(\displaystyle \langle a,b \rangle \in R\) and \(\displaystyle \langle b, a \rangle \in R \Longrightarrow \langle a, a \rangle \in R\) by transitivity ...But \(\displaystyle R\) is irreflexive ... so we have a contradiction ...Therefore \(\displaystyle \langle b, a \rangle \notin R\) ...
Can someone please critique my proof ... pointing out any errors or shortcomings ...
Such help will be much appreciated ...

Peter
 

1. What is a partial order relation?

A partial order relation is a mathematical concept that describes a relationship between two elements in a set. It is a binary relation that is reflexive, antisymmetric, and transitive. In simpler terms, it means that the elements in the set can be compared to each other and have some sort of order, but not all elements can be compared.

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

A total order relation is a binary relation that is reflexive, antisymmetric, transitive, and also complete, meaning that all elements in the set can be compared to each other. In contrast, a partial order relation does not have this completeness property, so there may be elements that cannot be compared to each other.

3. What are some real-life examples of partial order relations?

Some common examples of partial order relations in real life include ranking systems, such as the ranking of sports teams or universities. Another example is a job hierarchy, where some positions may be higher or lower than others, but not all positions can be directly compared to each other.

4. How are partial order relations represented?

Partial order relations can be represented in various ways, such as using diagrams or matrices. In a diagram, the elements are represented as points, and the relationships between them are shown with arrows. In a matrix, the elements are listed in rows and columns, and the presence of a 1 or 0 in each cell indicates whether there is a relationship between the elements or not.

5. What are the practical applications of partial order relations?

Partial order relations have various practical applications in fields such as computer science, economics, and social sciences. In computer science, they are used in sorting algorithms and data structures. In economics, they are used to model preferences and decision making. In social sciences, they are used to study hierarchies and power dynamics.

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