Well-Order

A well-order (in English: well-order) is a special case of total order. A set is well-ordered with respect to a given relation if the relation is a total order, AND every non-empty subset has a least element.

Formal Definition

Let (A, R) be an ordered set where R is a total order. The well-order condition:

This states that in any non-empty subset, we find an element such that there is no smaller one within that subset – i.e., there is a minimal element.

Examples of Well-Orders

• Natural numbers with ≤: every non-empty subset has a least element (induction principle).
• Positive integers with usual order: similar to natural numbers.
• Finite sets with total order: always well-ordered.

Counterexamples (Non-Well-Orders)

• Integers with ≤: the set of negative integers has no least element.
• Real numbers with ≤: the open interval (0,1) has no least element.
• Rational numbers with ≤: similar issue as reals.

Connection to Total Order

Every well-order is a total order, but not every total order is a well-order. The difference is that in well-orders, every subset has a minimal element. This property makes well-order a stronger condition than total order.

Summary

Well-order is one of the strongest types of ordering. Not only does it give a linear sequence to the set's elements, but it also ensures that every subset has a least element. This concept is key in many areas of mathematics, such as number theory and set theory.

Practice Exercise