Exercise · ZW-WELL-ORDERING-009 · ★★★ Advanced
Let \(A\) be a well-ordered set and \(f:A\to A\) such that \(f(a)<a\) for all \(a\in A\). What follows?
Choose one option.
Show answer
- C. That \(A\) is empty.
Explanation
If \(A\neq\varnothing\), its least element \(m\) would satisfy \(f(m)<m\) with \(f(m)\in A\), a contradiction; hence \(A\) is empty.
Review the concept: Well-ordering
A set \(A\) is well-ordered by \(<\) if \(<\) is a total order on \(A\) and every nonempty subset of \(A\) has a least element.
\[\forall S\subseteq A(S\ne\varnothing\Rightarrow\exists m\in S\forall s\in S\,m\le s)\]
More practice problems: well-ordering
- ZW-WELL-ORDERING-001 Which of the following statements describes a well-ordered set?
- ZW-WELL-ORDERING-002 Is ℤ well-ordered by its usual order?
- ZW-WELL-ORDERING-003 Which of these sets is well-ordered by the usual order of ℝ?
- ZW-WELL-ORDERING-004 If A is well-ordered and B⊆ A is nonempty, what holds?
- ZW-WELL-ORDERING-005 Let A=\2,5,7\ with the usual order. What is its least element, and why is A well-ordered?
- ZW-WELL-ORDERING-006 Is the set \x∈ℚ : x>0\ well-ordered by its usual order?
- ZW-WELL-ORDERING-007 Let B=\n∈ℕ : n^2>20\ with the usual order. What is its least element?
- ZW-WELL-ORDERING-008 Which of these subsets of ℤ has a least element?