WordNet

separated or at a distance in place or position or time; "These towns are many miles apart"; "stood with his legs apart"; "born two years apart"
not taken into account or excluded from consideration; "these problems apart, the country is doing well"; "all joking aside, I think youre crazy" (同)aside
into parts or pieces; "he took his fathers watch apart"; "split apart"; "torn asunder" (同)asunder
remote and separate physically or socially; "existed over the centuries as a world apart"; "preserved because they inhabited a place apart"- W.H.Hudson; "tiny isolated villages remote from centers of civilization"; "an obscure village" (同)isolated, obscure
away from another or others; "they grew apart over the years"; "kept apart from the group out of shyness"; "decided to live apart"
having characteristics not shared by others; "scientists felt they were a group apart"- Vannever Bush
one from the other; "people cant tell the twins apart"
not controlled by a party or interest group
(of a clause) capable of standing syntactically alone as a complete sentence; "the main (or independent) clause in a complex sentence has at least a subject and a verb" (同)main
free from external control and constraint; "an independent mind"; "a series of independent judgments"; "fiercely independent individualism"
to a remote degree; "it is remotely possible"
in a remote manner; "when the measured speech of the chorus passes over into song the tones are, remotely but unmistakably, those taught by the orthodox liturgy"
from or at a distance; "dimly, distantly, voices sounded in the stillness"
a suite of rooms usually on one floor of an apartment house (同)flat
racial segregation enforced primarily in public transportation and hotels and restaurants and other public places
a building that is divided into apartments (同)apartment_house
an apartment in a building without an elevator (同)walk-up
South American armadillo with three bands of bony plates (同)three-banded armadillo, Tolypeutes tricinctus

PrepTutorEJDIC

(時間的・空間的に)『離れて』,隔って;(…から)離れて《+『from』+『名』》 / 区別して;(…と)別々に《+『from』+『名』》 / ばらばらに
(他に)『頼らない』,依存しない / 『独立した』ほかの支配(影響)を受けない / 『関係(関連)がない』,独自の / (収入・資産が)働かなくても暮らせるだけの / 独立者,独立したもの / 党派に属さない人(無所属議員,自由な投票者など)
遠く離れて,遠い所で / よそよそしく
《米》『アパート』,(共同住宅内の)1家族分の[数]部屋 / 《複数形で》《英》(通例家具つきで貸す)一組の部屋 / 〈C〉部屋(room)

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/05/25 22:31:02」(JST)

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In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as # to distinguish from the negation of equality (the denial inequality) ≠, which is weaker.

Description

An apartness relation is a symmetric irreflexive binary relation with the additional condition that if two elements are apart, then any other element is apart from at least one of them (this last property is often called co-transitivity or comparison).

That is, a binary relation # is an apartness relation if it satisfies:

The negation of an apartness relation is an equivalence relation, as the above three conditions become reflexivity, symmetry, and transitivity. If this equivalence relation is in fact equality, then the apartness relation is called tight. That is, # is a tight apartness relation if it additionally satisfies:

4.

In classical mathematics, it also follows that every apartness relation is the negation of an equivalence relation, and the only tight apartness relation on a given set is the negation of equality. So in that domain, the concept is not useful. In constructive mathematics, however, this is not the case.

The prototypical apartness relation is that of the real numbers: two real numbers are said to be apart if there exists (one can construct) a rational number between them. In other words, real numbers x and y are apart if there exists a rational number z such that x < z < y or y < z < x. The natural apartness relation of the real numbers is then the disjunction of its natural pseudo-order. The complex numbers, real vector spaces, and indeed any metric space then naturally inherit the apartness relation of the real numbers, even though they do not come equipped with any natural ordering.

If there is no rational number between two real numbers, then the two real numbers are equal. Classically, then, if two real numbers are not equal, one would conclude that there exists a rational number between them. However it does not follow that one can actually construct such a number. Thus to say two real numbers are apart is a stronger statement, constructively, than to say that they are not equal, and while equality of real numbers is definable in terms of their apartness, the apartness of real numbers cannot be defined in terms of their equality. For this reason, in constructive topology especially, the apartness relation over a set is often taken as primitive, and equality is a defined relation.

A set endowed with an apartness relation is known as a constructive setoid. A function where A and B are constructive setoids is called a morphism for #_A and #_B if .

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