What (who) is well-ordered - definition

TOTAL ORDER SUCH THAT EVERY NONEMPTY SUBSET OF THE DOMAIN HAS A LEAST ELEMENT

Well-ordered set; Well-ordered; Well-ordering; Well ordered; Well ordering; Well-ordering property; Wellorder; Wellordering; Well ordered set; Wellordered; Well ordering theory; Well ordering property; Well-Ordering; Well-Ordered; Well-orderable set; Well order

well-ordered set

<mathematics> A set with a total ordering and no infinite descending chains. A total ordering "<=" satisfies x <= x x <= y <= z => x <= z x <= y <= x => x = y for all x, y: x <= y or y <= x In addition, if a set W is well-ordered then all non-empty subsets A of W have a least element, i.e. there exists x in A such that for all y in A, x <= y. Ordinals are isomorphism classes of well-ordered sets, just as integers are isomorphism classes of finite sets. (1995-04-19)

Well-quasi-ordering

$'''Pic.3:''' Hasse diagram of <math>\N^2</math> with componentwise order$

PREORDER IN WHICH EVERY INFINITE SEQUENCE HAS AN INCREASING OR EQUIVALENT PAIR OF CONSECUTIVE VALUES

Well partial order; WQO; Well quasi ordering; Wellquasiorder; Well-quasi-order; Well quasi order; Wqo; Well-quasi order; Well-partial-order

In mathematics, specifically order theory, a well-quasi-ordering or wqo is a quasi-ordering such that any infinite sequence of elements x_0, x_1, x_2, \ldots from X contains an increasing pair x_i\le x_j with i.

https://en.wikipedia.org/wiki/Well-quasi-ordering

All's Well That Ends Well

PLAY BY SHAKESPEARE

All's Well that Ends Well; All's well that ends well; Capilet; Parolles; All's well that ends well (proverb); Alls Well That Ends Well; All's Well That End's Well; All's Well, that Ends Well

All's Well That Ends Well is a play by William Shakespeare, published in the First Folio in 1623, where it is listed among the comedies. There is a debate regarding the dating of the composition of the play, with possible dates ranging from 1598 to 1608.

https://en.wikipedia.org/wiki/All's_Well_That_Ends_Well

Wikipedia

Well-order

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order 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 well-order relation is then called a well-ordered set. In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering.

Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see § Natural numbers below for an example). A well-ordered set S contains for every subset T with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of T in S.

If ≤ is a non-strict well ordering, then < is a strict well ordering. A relation is a strict well ordering if and only if it is a well-founded strict total order. The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible.

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well ordered. If a set is well ordered (or even if it merely admits a well-founded relation), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.

The observation that the natural numbers are well ordered by the usual less-than relation is commonly called the well-ordering principle (for natural numbers).

https://en.wikipedia.org/wiki/Well-order

Pronunciation examples for well-ordered

1. is a very well-ordered mechanism for

ted-talks_1827_StuartFirestein_2013-320k

2. And of course, a well-ordered life

ted-talks_526_MichaelMerzenich_2004-320k

3. So in my view, this was a very, very well-ordered,

Origins of Settled Life _ Ian Hodder _ Talks at Google

4. >>male #4: I was gonna say if-if there's runner up behind Getty Images for like well ordered

The Warhol Economy _ Elizabeth Currid-Halkett _ Talks at Google

5. So you said that Getty was like just comprehensive, well ordered and so on; TMZ chaotic is the

The Warhol Economy _ Elizabeth Currid-Halkett _ Talks at Google

Examples of use of well-ordered

1. Yet the shock in comfortable, well–ordered Britain is profound.

2. Now their well–ordered graves are adorned with plastic flowers, flags and personal mementoes.

3. A seemingly well–adjusted man in a well ordered universe is brought to the brink.

4. Savonarola knew that the republic had to be well–ordered, but he confused politics with religion and misjudgment with sin.

5. In well–ordered states, however, the media protects freedom by directing most of its firepower at those who rule.