WordNet

highly unusual or rare but not the single instance; "spoke with a unique accent"; "had unique ability in raising funds"; "a frankness unique in literature"; "a unique dining experience"
(followed by `to'
recurring only at long intervals; "a rare appearance"; "total eclipses are rare events"
having low density; "rare gasses"; "lightheaded from the rarefied mountain air" (同)rarefied, rarified
marked by an uncommon quality; especially superlative or extreme of its kind; "what is so rare as a day in June"-J.R.Lowell; "a rare skill"; "an uncommon sense of humor"; "she was kind to an uncommon degree" (同)uncommon
(of meat) cooked a short time; still red inside; "rare roast beef"
not widely distributed; "rare herbs"; "rare patches of green in the desert"
not widely known; especially valued for its uncommonness; "a rare word"; "rare books"
with nevertheless the final result; "He arrived only to find his wife dead"; "We won only to lose again in the next round"
never except when; "call me only if your cold gets worse" (同)only if, only when
as recently as; "I spoke to him only an hour ago"
except that; "It was the same story; only this time she came out better"
in the final outcome; "These news will only make you more upset"
typical or distinctive; "heard my friends characteristic laugh"; "red and gold are the characteristic colors of autumn"; "stripes characteristic of the zebra"
any measurable property of a device measured under closely specified conditions (同)device characteristic
a distinguishing quality
the integer part (positive or negative) of the representation of a logarithm; in the expression log 643 = 2.808 the characteristic is 2
grammatical number category referring to a single item or unit
the form of a word that is used to denote a singleton (同)singular form
the single one of its kind; "a singular example"; "the unique existing example of Donnes handwriting"; "a unique copy of an ancient manuscript"; "certain types of problems have unique solutions" (同)unique
being a single and separate person or thing; "can the singular person be understood apart from his culture?"; "every fact in the world might be singular...unlike any other fact and sole of its kind"-William James
composed of one member, set, or kind
put a new sole on; "sole the shoes" (同)resole
right-eyed flatfish; many are valued as food; most common in warm seas especially European
the underside of footwear or a golf club
lean flesh of any of several flatfish (同)fillet of sole
the underside of the foot

PrepTutorEJDIC

『ただ一つしかない』,唯一[無二]の / 『比類のない』,無比の,卓絶した / 《話》『珍しい』(rare);独特の,特異な
『まれな』,『珍しい』,めったにない / (空気などが)薄い,希薄な(thin) / 《話》すてきな,たいへんな
(肉などが)生焼けの,半煮えの
『唯一の』,たった一つ(一人)の;《複数名詞に付けて》ただ…だけ / 『最適の』,最良の(best) / 《数量に関して》『たった』,わずか / 『ただ…だけ』(『ばかり』),単に…に過ぎない / ただし,だが / 《英》《thatを伴って》…ということさえなければ
『特有の』,独特の / 『特性』,特色,特微
『非凡な』,目ざましい / 『奇妙な』,変わった / (文法で)『単数の』 / 単数[形]
足の裏 / 靴底 / 〈靴〉‘に'底をつける
『たった一人』(『一つ』)『の』 / 『独占的な』,一手に扱う
ウシノシタ,シタビラメ(ササウシノシタ科,ウシノシタ科の魚の総称)
公式発表,公報

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/02/16 22:15:56」(JST)

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[Wiki en表示]

"Unique" redirects here. For other uses, see Unique (disambiguation).

In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists. In mathematical logic, this sort of quantification is known as uniqueness quantification or unique existential quantification.

Uniqueness quantification is often denoted with the symbols "∃!" or ∃₌₁". For example, the formal statement

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

Proving uniqueness[edit]

The most common technique to proving uniqueness is to first prove existence of entity with the desired condition; then, to assume there exist two entities (say, a and b) that should both satisfy the condition, and logically deduce their equality, i.e. a = b.

As a simple high school example, to show x + 2 = 5 has only one solution, we assume there are two solutions first, namely, a and b, satisfying x + 2 = 5. Thus

By transitivity of equality,

By cancellation,

This simple example shows how a proof of uniqueness is done, the end result being the equality of the two quantities that satisfy the condition. We must say, however, that existence/expressibility must be proven before uniqueness, or else we cannot even assume the existence of those two quantities to begin with.

Reduction to ordinary existential and universal quantification[edit]

Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic by defining the formula ∃!x P(x) to mean literally,

which is the same as,

An equivalent definition that has the virtue of separating the notions of existence and uniqueness into two clauses, at the expense of brevity, is

Another equivalent definition with the advantage of brevity is

Generalizations[edit]

One generalization of uniqueness quantification is counting quantification. This includes both quantification of the form "exactly k objects exist such that …" as well as "infinitely many objects exist such that …" and "only finitely many objects exist such that…". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic.

References[edit]

Kleene, Stephen (1952). Introduction to Metamathematics. Ishi Press International. p. 199.
Andrews, Peter B. (2002). An introduction to mathematical logic and type theory to truth through proof (2. ed. ed.). Dordrecht: Kluwer Acad. Publ. p. 233. ISBN 1-4020-0763-9.

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