EDB — 0DQ

Exercises

1. [0DQ]Difficulty:*.(Proposed on 2022-12) Let the ordered set \((X,≤)\) be given; we define

   \[ P_ x {\stackrel{.}{=}}\{ w∈ X: w {\lt} x\} \quad . \]

   Suppose \((X,≤)\) meets these two requirements:

   • \[ ∀ x,y∈ X~ ,~ P_ x = P_ y ⇒ x=y \]

   • every non-empty set \(A⊆ X\) contains at least one minimal element, i.e.

     \[ ∃ a∈ A, ∀ b∈ A ¬(b{\lt}a)\quad ; \]

   then \((X,≤)\) is well ordered.

   Solution 1

   [26R]↺↻