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Formally, a partially ordered set ( P, ≤) is bounded complete if the following holds for any subset S of P:

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## Some Related Sentences

Formally and partially

__Formally__

**,**given

**a**

__partially__

**ordered**

**set**

**(**

**P**

**,**≤), then an element g

**of**

**a**

**subset**

**S**

**of**

**P**

**is**

**the**greatest element

**of**

**S**

**if**

__Formally__

**,**given two

__partially__

**ordered**sets

**(**

**S**

**,**

**≤)**and

**(**T

**,**≤),

**a**function f

**:**

**S**→ T

**is**an order-embedding

**if**f

**is**both order-preserving and order-reflecting

**,**i. e.

**for**all x and y in

**S**

**,**one has

__Formally__

**,**let

**P**be

**a**poset

**(**

__partially__

**ordered**

**set**), and let F be

**a**filter on

**P**; that

**is**

**,**F

**is**

**a**

**subset**

**of**

**P**such that

**:**

Formally and ordered

Formally and set

__Formally__

**,**

**the**

__set__

**of**all context-free languages

**is**identical to

**the**

__set__

**of**languages accepted by pushdown automata

**(**PDA ).

__Formally__speaking

**,**

**a**collation method typically defines

**a**total order on

**a**

__set__

**of**possible identifiers

**,**called sort keys

**,**which consequently produces

**a**total preorder on

**the**

__set__

**of**items

**of**information

**(**items with

**the**same identifier are not placed in

**any**defined order ).

__Formally__

**,**

**if**M

**is**

**a**

__set__

**,**

**the**identity function f on M

**is**defined to be that function with domain and codomain M which satisfies

__Formally__

**,**according to

**the**Constitution

**,**citizens

**of**Turkmenistan have

**the**right to

__set__up political parties and other public associations

**,**acting within

**the**framework

**of**

**the**Constitution and laws

**,**and public associations and groups

**of**citizens have

**the**right to nominate their candidates in accordance with

**the**election law.

__Formally__

**,**

**the**convex hull may be defined as

**the**intersection

**of**all convex sets containing X or as

**the**

__set__

**of**all convex combinations

**of**points in X.

__Formally__

**,**

**the**discrete cosine transform

**is**

**a**linear

**,**invertible function

**(**where denotes

**the**

__set__

**of**real numbers ), or equivalently an invertible N × N square matrix.

__Formally__

**,**we are given

**a**

__set__

**of**hypotheses and

**a**

__set__

**of**manifestations ; they are related by

**the**domain knowledge

**,**represented by

**a**function that takes as an argument

**a**

__set__

**of**hypotheses and gives as

**a**result

**the**corresponding

__set__

**of**manifestations.

__Formally__

**,**

**a**function ƒ

**is**real analytic on an open

__set__D in

**the**real line

**if**

**for**

**any**x < sub > 0 </ sub > in D one can write

__Formally__

**,**

**the**movement

**is**

**a**rondo that acts as

**the**theme

**for**

**a**

__set__

**of**eight variations

**,**capped off by

**a**dramatic coda.

__Formally__

**,**

**the**discrete sine transform

**is**

**a**linear

**,**invertible function F

**:**R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup >

**(**where R denotes

**the**

__set__

**of**real numbers ), or equivalently an N × N square matrix.

__Formally__

**,**

**the**discrete Hartley transform

**is**

**a**linear

**,**invertible function H

**:**R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup >

**(**where R denotes

**the**

__set__

**of**real numbers ).

__Formally__

**,**

**a**constraint satisfaction problem

**is**defined as

**a**triple

**,**where

**is**

**a**

__set__

**of**variables

**,**

**is**

**a**domain

**of**values

**,**and

**is**

**a**

__set__

**of**constraints.

Formally and P

__Formally__

**,**

**a**decision problem

**is**P-complete

**(**

**complete**

**for**

**the**complexity class

__P__)

**if**it

**is**in

__P__and that every problem in

__P__can be reduced to it by using an appropriate reduction.

__Formally__

**,**define

**the**

**set**

**of**lines in

**the**plane

__P__as L

**(**

__P__); then

**a**rigid motion

**of**

**the**plane takes lines to lines –

**the**group

**of**rigid motions acts on

**the**

**set**

**of**lines – and one may ask which lines are unchanged by an action.

__Formally__

**,**

**a**product term

__P__in

**a**sum

**of**products

**is**an implicant

**of**

**the**Boolean function F

**if**

__P__implies F. More precisely

**:**

__Formally__

**,**

__P__

**is**

**a**symmetric polynomial

**,**

**if**

**for**

**any**permutation σ

**of**

**the**subscripts 1

**,**2

**,**..., n one has

__P__

**(**X < sub > σ

**(**1 )</ sub >, X < sub > σ

**(**2 )</ sub >, …, X < sub > σ

**(**n )</ sub >) =

__P__

**(**X < sub > 1 </ sub >, X < sub > 2 </ sub >, …, X < sub > n </ sub >).

Formally and is

__Formally__organized vocational programs supported by federal funds allow high school students to gain experience in

**a**field

**of**work which

__is__likely to lead to

**a**full-time job on graduation.

More rigorously

**,****the**divergence**of****a**vector field F at**a**point p__is__defined as**the**limit**of****the**net flow**of**F across**the**smooth boundary**of****a**three dimensional region V divided by**the**volume**of**V as V shrinks to p.__Formally__**,**__Formally__

**,**

**the**base

__is__known as Naval Support Facility Diego Garcia

**(**

**the**US activity ) or Permanent Joint Operating Base

**(**PJOB ) Diego Garcia

**(**

**the**UK's term ).

__Formally__

**,**there

__is__

**a**clear distinction

**:**" DFT " refers to

**a**mathematical transformation or function

**,**regardless

**of**how it

__is__computed

**,**whereas " FFT " refers to

**a**specific family

**of**algorithms

**for**computing DFTs.

__Formally__

**,**oxidation state

__is__

**the**hypothetical charge that an atom would have

**if**all bonds to atoms

**of**different elements were 100 % ionic.

__Formally__

**,**an inner product space

__is__

**a**vector space V over

**the**field together with an inner product

**,**i. e., with

**a**map

*

__Formally__**,**when working over**the**reals**,**as here**,**this__is__accomplished by considering**the**limit as ε → 0 ; but**the**" infinitesimal " language generalizes directly to Lie groups over general rings.__Formally__

**,**

**a**profinite group

__is__

**a**Hausdorff

**,**compact

**,**and totally disconnected topological group

**:**that

__is__

**,**

**a**topological group that

__is__also

**a**Stone space.

__Formally__

**,**this sharing

**of**dynamics

__is__referred to as universality

**,**and systems with precisely

**the**same critical exponents are said to belong to

**the**same universality class.

__Formally__

**,**

**a**frame

__is__defined to be

**a**lattice L in which finite meets distribute over arbitrary joins

**,**i. e. every

**(**even infinite )

**subset**

0.221 seconds.