currently I am working on an exercise, where I have to give the supremum of $\emptyset\subset A$. The set $A$ is defined as follows:
$A:=$ {{1},{1,2},{1,3,4},{1,2,3,4}}. On this set, we define the inclusion order.
In general, the supremum is the upperbound, which is an element of $A$. Is it true that the supremum is equal to {1}, since it is contained in every other element of $A$? Is it possible that the empty set is the supremum?
Thanks in advance!
Supremum would be the smallest set containing all the sets that you have. What you have written is the smallest contained in all the sets, which is the infimum of sets.
Notation wise, $$\sup_n A_n=\cup_{i=1}^{n}A_i\qquad\text{ and }\qquad\inf_nA_n=\cap_{i=1}^nA_i$$