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In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the ordering is then called a well-ordered set.
10 paź 2024 · Every finite totally ordered set is well ordered. The set of integers Z, which has no least element, is an example of a set that is not well ordered. An ordinal number is the order type of a well ordered set.
8 sty 2017 · A well-ordered set is a totally ordered set that satisfies the minimum condition. Learn the definition, examples, properties, and theorems of well-ordered sets and their relation to ordinal numbers.
A set \(T\) of real numbers is said to be well-ordered if every nonempty subset of \(T\) has a smallest element. Therefore, according to the principle of well-ordering, \(\mathbb{N}\) is well-ordered.
In contrast, a well-ordered set is a totally ordered set with an additional property that every subset $W$ of $P$ contains a smallest element $s\in W$ in the sense that for any $a\in W$ we have $s\le a$.
Learn the definition, uses, and proofs of the well-ordering principle, which states that every nonempty subset of positive integers has a least element. See how it is related to the principle of mathematical induction and the axiom of choice.
18 wrz 2023 · Two sets are equal if and only if they have the same elements. So {1, 2, 3} = {3, 2, 1}, etc. When we speak of a partially ordered set, what we are really referring to is a pair (S, R), where S is a set and R ⊆ S × S is a relation that satisfies the axioms of a partial ordering.
