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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