Strong or weak orders?
For the following relations on A, state whether they are strong or weak orders on A or neither.?
Given A={a,b,c,d}
For the following relations on A, indicate whether they are strong or weak orders on A or neither.
{⟨d,c⟩,⟨d,b⟩,⟨d,a⟩,⟨c,b⟩,⟨c,a⟩,⟨b,a⟩}
weak
{⟨b,a⟩,⟨c,b⟩,⟨c,a⟩}
neither nor
{⟨d,c⟩,⟨d,b⟩,⟨d,a⟩,⟨c,b⟩,⟨c,a⟩,⟨b,a⟩}
neither nor
{⟨a,b⟩,⟨a,c⟩,⟨a,d⟩,⟨b,c⟩,⟨a,a⟩,⟨b,b⟩,⟨c,c⟩,⟨d,d⟩,⟨c,b⟩}
strong
{⟨a,b⟩,⟨a,c⟩,⟨a,d⟩,⟨b,c⟩,⟨a,a⟩,⟨b,b⟩,⟨c,c⟩}
weak
{⟨d,c⟩,⟨d,b⟩,⟨d,a⟩,⟨c,a⟩,⟨a,b⟩}
neither nor
{⟨a,b⟩,⟨a,c⟩,⟨a,d⟩,⟨b,c⟩}
neither nor
{⟨b,a⟩,⟨b,b⟩,⟨a,a⟩,⟨c,c⟩,⟨d,d⟩,⟨c,b⟩,⟨c,a⟩}
neither nor
That's how I assigned it. Is that correct?
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What do you think? A strong order is not reflexive.
https://de.wikipedia.org/wiki/Ordnungsrelation#Strenge_Totalorganisation
Then I misunderstood that. What about the other examples? Are they as correct as possible?
Best regards and thank you
I'm not going through all this now, and I just don't have time.
{⟨d,c⟩,⟨d,b⟩d,a⟩,⟨c,b⟩c,a⟩,b⟩,a⟩,a⟩}
Why don't you write all around as
d < c
d < b
d <
c < b
c <
b <
Then you can sort it better. It is then true how to easily see the chain
d < c < b <
and with that you have a strong order.
All right! Could you tell me which one of them is a strong order so that I can compare one?