Family of sets without maximal element

[Bipolarity]

I have begun to learn about maximal elements from a linear algebraic perspective (maximal linearly independent subsets of vector spaces). I have a few questions of which I have been able to get few insights online:

1) Does every family of sets have a maximal element? How can I make a family of sets that does not have a maximal element? I have to obviously make the hypothesis of Zorn's lemma fail, but I can't quite see how to do that.

2) Does every chain of sets have a maximal element? It seems that a chain of sets necessarily satisfies the criteria for Zorn's lemma but I am not sure.

Thanks!

BiP

 

[HallsofIvy]

No, to both of those. A counter-example is $$A_n= [-n, n]$$, the closed interval from -n to n.

Zorns lemma says that if every chain in an ordered collection has a maximum, then the collection has maximal elements. If it were true that every chain has a maximum or that every family has a maximum, Zorn's lemma would be unnecessary.