詳細に明らかにされた

WordNet

accurately stated or described; "a set of well-defined values" (同)clear
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"
give a definition for the meaning of a word; "Define `sadness"
show the form or outline of; "The tree was clearly defined by the light"; "The camera could define the smallest object" (同)delineate
determine the nature of; "What defines a good wine?"
clearly characterized or delimited; "lost in a maze of words both defined and undefined"; "each child has clearly defined duties"
showing clearly the outline or profile or boundary; "hills defined against the evening sky"; "the setting sun showed the outlined figure of a man standing on the hill" (同)outlined

PrepTutorEJDIC

輪郭のはっきりした;(定義・意義が)明確な
『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
〈語・句〉'を'『定義する』,‘の'意味で明らかにする / 〈本質・意義・立場〉'を'『明らかにする』,特徴づける / …‘の'輪郭をはっきりさせる / …‘の'限界(範囲)を決める
『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）』「2015/06/18 19:48:17」(JST)

wiki ja

[Wiki ja表示]

well-defined は、ある概念が数学的あるいは論理学的に特定の条件を公理に用いて定義・導入されるとき、その定義（における公理の組）が自己矛盾をその中に含み持たぬ状態にあることを言い表す修飾語句である。また、ある概念の定義をする場合、そう決めることによって、何も論理的な矛盾なく上手くいくということ（定義の整合性）が確認されているということを言い表す言葉である。文脈により、「うまく定義されている」「矛盾なく定まった」「定義可能である」などと表現されることもある。

well-defined は「状態」を表す形容詞であるが、日本語の定訳はなく慣例的に形容詞と動詞の複合語に訳されるか、そのまま形容動詞的に「well-defined である」といった形で用いる。名詞形 well-definedness などもあり、これを well-defined 性と記すことはできるが日本語訳としてこなれたものは特には存在しない（文脈によっては「定義可能性」などで代用可能である）。

概要

以下の二つが示せたとき、定義が well-defined であるという^[1]。

(1) 定義で使われる方法が実際にうまくいく。

(2) 定義がもともとの対象から複数定まる対象を経由して行われる場合、結果がもともとの対象にのみ依存する。

一つの対象のある表示に対して定義が満たされるが、別のある表示については満たされない状況であるとか、一つの対象の異なる表示を考えると定義の示す結果がそれぞれの表示に対して異なるといった状況であるならば、与えられた定義はその対象自体に対する定義として不適切 (ill-defined) である。

例

例えば、写像あるいは（一価の）関数 f は代入原理と呼ばれる条件

を満たす対応（一意対応）でなければならないから、同値類に対する写像をその代表元を用いて定義しようとする場面などでは well-defined 性が問題になる。典型的なものが、代数学において商代数系（商群や商環、商ベクトル空間など）の演算を導入する場面に現れる。

鎖複体の射からホモロジー（これは鎖複体から定まるある商加群である）の間の準同型が誘導されるが、このときも well-defined 性が問題になる。上述の一意性に加え、写像の行き先が実際に終域に入っていることを確かめなくてはならない。

実数 a > 0 の実数 x 乗を、x に収束する有理数列 {x_n} を用いて

と定義するときにも、well-defined 性が問題になる^[1]。右辺の収束性と一意性（{x_n} の取り方によらないこと）である。

参考文献

^ ^a ^b 雪江明彦『代数学1 群論入門』日本評論社、2010年。ISBN 978-4-535-78659-2。

wiki en

[Wiki en表示]

In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well-defined or ambiguous.^[1] A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well-defined (and thus: not a function).^[2] The term well-defined is also used to indicate whether a logical statement is unambiguous.

Well-defined functions

Let be sets, let and define as if and if . Then is well-defined if . If however then is not well-defined because is ambiguous for .

In group theory, the term well-defined is often used when dealing with cosets, where a function f on a quotient group may be defined in terms of a coset representative. Here, a necessary requirement for f to be considered a function is that the output must be independent of which coset representative is chosen. The phrase f is well-defined is used to indicate that this requirement has been verified.

For example, consider , the integers modulo 2. Since 4 and 6 are congruent modulo 2, a function f whose domain is must give the same output when the input is represented by 4 that it gives when the input is represented by 6.

A function that is not well-defined is not the same as a function that is undefined. For example, if f(x) = 1/x, then f(0) is undefined, but this has nothing to do with the question of whether f(x) = 1/x is well-defined. It is; 0 is simply not in the domain of the function.

Operations

In particular, the term well-defined is used with respect to (binary) operations on cosets. In this case one can view the operation as a function of two variables and the property of being well-defined is the same as that for a function. For example, addition on the integers modulo some n can be defined naturally in terms of integer addition.

The fact that this is well-defined follows from the fact that we can write any representative of as , where k is an integer. Therefore,

and similarly for any representative of .

Well-defined notation

For real numbers, the product is unambiguous because .^[1] In this case this notation is said to be well-defined. However, if the operation (here ) did not have this property, which is known as associativity, then there must be a convention for which two elements to multiply first. Otherwise, the product is not well-defined. The subtraction operation, , is not associative, for instance. However, the notation is well-defined under the convention that the operation is understood as addition of the opposite, thus is the same as . Division is also non-associative. However, does not have an unambiguous conventional interpretation, so this expression is ill-defined.

Other uses of the term

A solution to a partial differential equation is said to be well-defined if it is determined by the boundary conditions in a continuous way as the boundary conditions are changed.^[1]

References

Notes

^ ^a ^b ^c Weisstein, Eric W. "Well-Defined". From MathWorld--A Wolfram Web Resource. Retrieved 2 January 2013.
^ Joseph J. Rotman, The Theory of Groups: an Introduction, p. 287 "... a function is "single-valued," or, as we prefer to say ... a function is well defined.", Allyn and Bacon, 1965.

Books

Contemporary Abstract Algebra, Joseph A. Gallian, 6th Edition, Houghlin Mifflin, 2006, ISBN 0-618-51471-6.
Algebra: Chapter 0, Paolo Aluffi, ISBN 978-0821847817. Page 16.
Abstract Algebra, Dummit and Foote, 3rd edition, ISBN 978-0471433347. Page 1.

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