R is a partial order relation on some set A, which can be either finite or infinite. Which of the following statements are true and which are false?
1) With this order, A has at least one minimal element and at least one maximal element.
2) If with this order, A has a smallest and largest elements, then every two element of A are comparable.
3) If with this order, A has no maximal elements, then A is infinite.
4) if with this order, A has a single minimal element, then it is a smallest element.
5) If every two elements are comparable, then there is a smallest and largest elements.
I have a dilemma regarding the largest element. Can I say, that in the set of all subsets of A, ordered by inclusion, A itself is "largest", and the empty set is "smallest"?
I so, does it means that P(N) has a largest element, which is N itself?
Regarding the statements, I think that 3 and 4 are true and the others are false, but I am not sure I am correct. Number 2 is especially confusing me because of the largest element. What do you think ?
Thank you
The subset relation is a partial order, and if you take the sets $\emptyset$, $\{a\}$, $\{b\}$, and $\{a,b\}$, then there is a smallest and largest element, but $\{a\}$ and $\{b\}$ are not comparable. So that would be a counterexample to 2.