WordNet

without unusual distress or resentment; with good humor; "took the joke well"; "took the tragic news well"
a cavity or vessel used to contain liquid
a deep hole or shaft dug or drilled to obtain water or oil or gas or brine
an enclosed compartment in a ship or plane for holding something as e.g. fish or a planes landing gear or for protecting something as e.g. a ships pumps
an open shaft through the floors of a building (as for a stairway)
an abundant source; "she was a well of information" (同)wellspring, fountainhead
in a manner affording benefit or advantage; "she married well"; "The children were settled advantageously in Seattle" (同)advantageously
in financial comfort; "They live well"; "she has been able to live comfortably since her husband died" (同)comfortably
to a great extent or degree; "Im afraid the film was well over budget"; "painting the room white made it seem considerably (or substantially) larger"; "the house has fallen considerably in value"; "the price went up substantially" (同)considerably, substantially
indicating high probability; in all likelihood; "I might well do it"; "a mistake that could easily have ended in disaster"; "you may well need your umbrella"; "he could equally well be trying to deceive us" (同)easily
(often used as a combining form) in a good or proper or satisfactory manner or to a high standard (`good is a nonstandard dialectal variant for `well' (同)good
with great or especially intimate knowledge; "we knew them well" (同)intimately
come up, as of a liquid; "Tears well in her eyes"; "the currents well up" (同)swell
(used for emphasis or as an intensifier) entirely or fully; "a book well worth reading"; "was well aware of the difficulties ahead"; "suspected only too well what might be going on"
favorably; with approval; "their neighbors spoke well of them"; "he thought well of the book"
in good health especially after having suffered illness or injury; "appears to be entirely well"; "the wound is nearly well"; "a well man"; "I think Im well; at least I feel well"
thoroughly or completely; fully; often used as a combining form; "The problem is well understood"; "she was well informed"; "shake well before using"; "in order to avoid food poisoning be sure the meat is well cooked"; "well-done beef", "well-satisfied customers"; "well-educated"
to a suitable or appropriate extent or degree; "the project was well underway"; "the fetus has well developed organs"; "his father was well pleased with his grades"
wise or advantageous and hence advisable; "it would be well to start early"
with prudence or propriety; "You would do well to say nothing more"; "could not well refuse"
with skill or in a pleasing manner; "she dances well"; "he writes well"
come upon unexpectedly or after searching; "found art"; "the lost-and-found department"
food and lodging provided in addition to money; "they worked for $30 and found"

PrepTutorEJDIC

事実に基づいた,根拠の確かな
『井戸』 / (石油・天然ガスを取るための)井 / 『泉』,《比喩的に》源 / 井戸状のくぼみ;(エレベーターの)縦穴;(回り階段の)階段穴 / 《英》(法廷の)弁護士席 / 〈水などが〉わき出る,噴出する《+out》 / わき上がる《+up》
『よく』,満足に,申し分なく / じょうずに,うまく,巧みに / 『十分に』,全く / 《他の副詞または前置詞句と共に用いて》『相当に』,かなり / (個人的に)親しく,親密に / 《can't, couldn't と共に用いて》『正当に』,適切に / はっきりと,確かに,全く / 親切に,好意的に / 《おもに補語に用いて》『健康な』,じょうぶな / 《複語にのみ用いて》『好都合の』,満足すべき,申し分のない / 《驚いて》『おや』,まあ,へ‐ / 《ためらい・同意を表し,文・言葉を切り出して》『え‐と』,さて,ところで / 《譲歩を表して》『それじゃ』,まあ / 《安心・あきらめを表して》『やれやれ』,どうも
findの過去・過去分詞
…‘を'『創設する』,設立する(establish) / (…の上に)〈建物など〉‘を'建てる;(…に)…‘の'基礎を置く《+『名』+『on』(『upon』)+『名』(do『ing』)》
〈金属・ガラス〉を溶かして型に入れる / 〈物〉‘を'鋳造する
『私たちは(が)』,われわれは(が) / 《総称的に》『人は』,われわれは / 《論説・評論などで I の代りに用いて》私たちは,吾人(ごじん)は,筆者は / 《子供・病人などに対して,you の代りに養いて》あなたは / 《君主が公式の場でI の代りに用いて》余は

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/07/21 10:42:51」(JST)

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In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-empty subset S ⊆ X has a minimal element; that is, some element m not related by sRm (for instance, "m is not smaller than s") for any s ∈ S.

\forall S\subseteq X\ (S\neq \varnothing \to \exists m\in S\;\;\forall s\in S\;\,(s,m)\notin R).

(Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.)

Equivalently, assuming some choice, a relation is well-founded if it contains no countable infinite descending chains: that is, there is no infinite sequence x₀, x₁, x₂, ... of elements of X such that x_n+1 R x_n for every natural number n.

In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order.

In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all sets are well-founded.

A relation R is converse well-founded, upwards well-founded or Noetherian on X, if the converse relation R⁻¹ is well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of rewriting systems, a Noetherian relation is also called terminating.

Induction and recursion

An important reason that well-founded relations are interesting is because a version of transfinite induction can be used on them: if (X, R) is a well-founded relation, P(x) is some property of elements of X, and we want to show that

P(x) holds for all elements x of X,

it suffices to show that:

If x is an element of X and P(y) is true for all y such that y R x, then P(x) must also be true.

That is,

\forall x\in X\,[(\forall y\in X\,(y\,R\,x\to P(y)))\to P(x)]\to \forall x\in X\,P(x).

Well-founded induction is sometimes called Noetherian induction,^[1] after Emmy Noether.

On par with induction, well-founded relations also support construction of objects by transfinite recursion. Let (X, R) be a set-like well-founded relation and F a function that assigns an object F(x, g) to each pair of an element x ∈ X and a function g on the initial segment {y: y R x} of X. Then there is a unique function G such that for every x ∈ X,

G(x)=F(x,G\vert _{\{y:y\,R\,x\}})

That is, if we want to construct a function G on X, we may define G(x) using the values of G(y) for y R x.

As an example, consider the well-founded relation (N, S), where N is the set of all natural numbers, and S is the graph of the successor function x → x + 1. Then induction on S is the usual mathematical induction, and recursion on S gives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-values recursion. The statement that (N, <) is well-founded is also known as the well-ordering principle.

There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction. When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction. See those articles for more details.

Examples

Well-founded relations which are not totally ordered include:

the positive integers {1, 2, 3, ...}, with the order defined by a < b if and only if a divides b and a ≠ b.
the set of all finite strings over a fixed alphabet, with the order defined by s < t if and only if s is a proper substring of t.
the set N × N of pairs of natural numbers, ordered by (n₁, n₂) < (m₁, m₂) if and only if n₁ < m₁ and n₂ < m₂.
the set of all regular expressions over a fixed alphabet, with the order defined by s < t if and only if s is a proper subexpression of t.
any class whose elements are sets, with the relation $\in$ ("is an element of"). This is the axiom of regularity.
the nodes of any finite directed acyclic graph, with the relation R defined such that a R b if and only if there is an edge from a to b.

Examples of relations that are not well-founded include:

the negative integers {-1, -2, -3, …}, with the usual order, since any unbounded subset has no least element.
The set of strings over a finite alphabet with more than one element, under the usual (lexicographic) order, since the sequence "B" > "AB" > "AAB" > "AAAB" > … is an infinite descending chain. This relation fails to be well-founded even though the entire set has a minimum element, namely the empty string.
the rational numbers (or reals) under the standard ordering, since, for example, the set of positive rationals (or reals) lacks a minimum.

Other properties

If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n − 1, n − 2, ..., 2, 1 has length n for any n.

The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X,R) is isomorphic to (C,∈).

Reflexivity

A relation R is said to be reflexive if a R a holds for every a in the domain of the relation. Every reflexive relation on a nonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example, in the natural numbers with their usual order ≤, we have $1\geq 1\geq 1\geq \cdots$ . To avoid these trivial descending sequences, when working with a reflexive relation R it is common to use (perhaps implicitly) the alternate relation R′ defined such that a R′ b if and only if a R b and a ≠ b. In the context of the natural numbers, this means that the relation <, which is well-founded, is used instead of the relation ≤, which is not. In some texts, the definition of a well-founded relation is changed from the definition above to include this convention.

References

^ Bourbaki, N. (1972) Elements of mathematics. Commutative algebra, Addison-Wesley.

Just, Winfried and Weese, Martin, Discovering Modern Set theory. I, American Mathematical Society (1998) ISBN 0-8218-0266-6.

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