My background in set relations is from Velleman's How to Prove It, which I read admittedly a few weeks ago. I'm unfamiliar with certain things presented in Schmidt and Ströhlein's book "Relations and Graphs."
Firstly, I am likely misunderstanding that it stating that the supremum (l.u.b.) and infimum (g.l.b.) as existing for an arbitrary set. However, aren't there conditions for this, as described in the below screenshot? (The first screenshot is from Velleman and the second from Relations and Graphs).
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Secondly, I don't understand the logic behind transpositions and the complements of sup/inf for a given relation.
