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

[Math Amateur]

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

 

[Math Amateur]

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 $$R$$ ... $$\langle a,b \rangle \in R \Longrightarrow \langle b, a \rangle \notin R$$ ...So ... my attempt at a proof is as follows:Assume $$\langle a,b \rangle \in R$$ ...NOW ... ALSO ASSUME that $$\langle b, a \rangle \in R$$ ... and look for a contradiction ...... so ... now we have ...$$\langle a,b \rangle \in R$$ and $$\langle b, a \rangle \in R \Longrightarrow \langle a, a \rangle \in R$$ by transitivity ...But $$R$$ is irreflexive ... so we have a contradiction ...Therefore $$\langle b, a \rangle \notin R$$ ...
Can someone please critique my proof ... pointing out any errors or shortcomings ...
Such help will be much appreciated ...

Peter