Exercise · ZW-WELL-ORDERING-005 · ★★☆ Intermediate

Let \(A=\{2,5,7\}\) with the usual order. What is its least element, and why is \(A\) well-ordered?

Choose one option.

Show answer

C. The least element is \(2\); since \(A\) is finite, every nonempty subset has a least element.

Explanation

Since \(A\) is finite and totally ordered, every nonempty subset has a least element, and the smallest of its elements is \(2\).

A set \(A\) is well-ordered by \(<\) if \(<\) is a total order on \(A\) and every nonempty subset of \(A\) has a least element.

\[\forall S\subseteq A(S\ne\varnothing\Rightarrow\exists m\in S\forall s\in S\,m\le s)\]