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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

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  • $\begingroup$ Only 3 is correct. Can you find a counter example to 4? $\endgroup$ – William Elliot Jun 5 '17 at 5:27
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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.

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  • $\begingroup$ I see, so if I can't determine which set is a subset of which set, in the case of {a} and {b} they are not comparable? I understand why the subset relation is a partial order, but when the set is infinite, like the natural numbers, is it still correct to say it has a largest element, which is N itself? $\endgroup$ – user3275222 Jun 5 '17 at 6:06
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    $\begingroup$ @user3275222 Correct! To be precise, when you use the subset relation on the power set of the natural numbers, then the set of all natural numbers is a (and the only one) largest element. (And of course the empty set would be a (and the only one) smallest element. $\endgroup$ – Bram28 Jun 5 '17 at 10:59

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