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
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