Exercise · ZW-ZORN-S-LEMMA-009 · ★★☆ Intermediate
What is the upper bound of the chain \(\{\{1,\dots,n\}: n\in\mathbb{N}\}\) among the finite subsets of \(\mathbb{N}\)?
Choose one option.
A The set \(\{1\}\). B The set \(\mathbb{N}\), which is finite. C The empty set. D No upper bound among the finite subsets of \(\mathbb{N}\). Show answer D. No upper bound among the finite subsets of \(\mathbb{N}\).← Back to the question
Explanation The union of the chain is \(\mathbb{N}\), which is an upper bound but does not belong to the family of finite sets .
Zorn's lemma is a principle equivalent to the axiom of choice stating that every nonempty partially ordered set in which every chain has an upper bound contains a maximal element.
\[\\forall P\\neq\\varnothing\\,\\Big[\\Big(\\forall C\\subseteq P\\,\\big(\\text{cadena}(C)\\Rightarrow\\\\\exists u\\in P\\,\\forall c\\in C\\,c\\le u\\big)\\Big)\\Rightarrow\\\\\exists m\\in P\\ \\text{maximal}\\Big]\]
More practice problems: zorn's lemma