Exercise · ZW-ZORN-S-LEMMA-008 · ★★☆ Intermediate
Let \(P=\{1,2,3,\dots\}\) with the divisibility order. Does \(P\) have a maximal element?
Choose one option.
Show answer
- A. No, no number is maximal.
Explanation
For any \(n\) there is \(2n\) with \(n\mid 2n\), so no element is maximal.
Review the concept: Zorn's lemma
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
- ZW-ZORN-S-LEMMA-001 What does Zorn's lemma state?
- ZW-ZORN-S-LEMMA-002 Which axiom is Zorn's lemma equivalent to?
- ZW-ZORN-S-LEMMA-003 What is a chain in a partially ordered set?
- ZW-ZORN-S-LEMMA-004 What does it mean for m to be a maximal element of P?
- ZW-ZORN-S-LEMMA-005 What condition must every chain satisfy to apply Zorn's lemma?
- ZW-ZORN-S-LEMMA-006 In P(\1,2\) ordered by inclusion, how many maximal elements are there?
- ZW-ZORN-S-LEMMA-007 Let P=\\1\,\2\,\1,2\\ ordered by inclusion. What is a maximal chain of P?
- ZW-ZORN-S-LEMMA-009 What is the upper bound of the chain \\1,dots,n\: n∈ℕ\ among the finite subsets of ℕ?