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Formally, given a finite set X, a collection C of subsets of X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z.

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## Some Related Sentences

Formally and given

__Formally__

__given__to bishop Philip

**of**Senj

**,**the permission to use the Glagolitic liturgy ( the Roman Rite conducted

**in**Slavic language instead

**of**Latin

**,**not the Byzantine rite ), actually extended to

**all**Croatian lands

**,**mostly along the Adriatic coast

**.**

__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__

**,**the statement

**that**" value decreases over time " is

__given__by defining the linear differential operator as:

__Formally__

**,**it is

**a**norm defined on the space

**of**bounded linear operators between

**two**

__given__normed vector spaces

**.**

__Formally__

**,**the problem

**can**be stated as follows:

__given__

**a**desired property

**,**expressed as

**a**temporal logic formula p

**,**

**and**

**a**structure M with initial state s

**,**decide

**if**

**.**

__Formally__

**,**

__given__

**two**categories

**C**

**and**D

**,**an equivalence

**of**categories consists

**of**

**a**functor F:

**C**→ D

**,**

**a**functor G: D →

**C**

**,**

**and**

**two**natural isomorphisms ε: FG → I < sub > D </ sub >

**and**η: I < sub >

**C**</ sub >→ GF

**.**

__Formally__

**,**an absolute coequalizer

**of**

**a**pair

**in**

**a**category

**C**is

**a**coequalizer as defined above but with the added property

**that**

__given__any functor F ( Q ) together with F ( q ) is the coequalizer

**of**F ( f )

**and**F ( g )

**in**the category D

**.**Split coequalizers are examples

**of**absolute coequalizers

**.**

__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__

**,**

**a**deterministic algorithm computes

**a**mathematical function ;

**a**function

**has**

**a**unique value for any

__given__input

**,**

**and**the algorithm is

**a**process

**that**produces this particular value as output

**.**

__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__

**,**

__given__

**a**graph G

**,**

**a**vertex labeling is

**a**function mapping vertices

**of**G to

**a**

**set**

**of**labels

**.**

__Formally__this mission was declared to be only

**in**pursuit

**of**the Russian vessels

**and**ammunition taken to Anzali by the White Russian counter-revolutionary General Denikin

**,**who had been

__given__asylum by British forces

**in**Anzali

**.**

__Formally__

**,**

**a**married or widowed woman

**can**be called by the

__given__name

**of**her husband ( Madame (

__given__name

**of**husband ) family name or Madame veuve (

__given__name

**of**husband ) family name ); this is now slightly out

**of**fashion

**.**

Formally and finite

__Formally__

**,**

**a**frame is defined to be

**a**lattice L

**in**which

__finite__

**meets**distribute over arbitrary joins

**,**i

**.**e

**.**

**every**( even infinite ) subset

__Formally__

**,**

**we**start with

**a**category

**C**with

__finite__products ( i

**.**e

**.**

**C**

**has**

**a**terminal object 1

**and**any

**two**objects

**of**

**C**have

**a**product ).

__Formally__

**,**the question

**of**whether the universe is infinite or

__finite__is whether it is an unbounded or bounded metric space

**.**

__Formally__

**,**an iterated function system is

**a**

__finite__

**set**

**of**contraction mappings on

**a**complete metric space

**.**

__Formally__

**,**the game ( coalitional game ) consists

**of**

**a**

__finite__

**set**

**of**players

**,**called the grand coalition

**,**

**and**

**a**characteristic function denotes the power

**set**

**of**.</ ref > from the

**set**

**of**

**all**possible coalitions

**of**players to

**a**

**set**

**of**payments

**that**satisfies

**.**

__Formally__

**,**

**a**system is said to be observable

**if**

**,**for any possible sequence

**of**state

**and**control vectors

**,**the current state

**can**be determined

**in**

__finite__time using only the outputs ( this definition is slanted towards the state space representation ).

__Formally__

**,**

**we**define

**a**bad field as

**a**structure

**of**the form ( K

**,**T ), where K is an algebraically closed field

**and**T is an infinite

**,**proper

**,**distinguished subgroup

**of**K

**,**

**such**

**that**( K

**,**T ) is

**of**

__finite__Morley rank

**in**its full language

**.**

__Formally__

**,**

**a**deterministic

__finite__automaton may be defined by the tuple where is the

**set**

**of**states

**of**the automaton

**,**is the

**set**

**of**input symbols

**,**is the transition function

**that**takes

**a**state

**and**an input symbol to

**a**new state

**,**is the initial state

**of**the automaton

**,**

**and**is the

**set**

**of**accepting or final states

**of**the automaton

**.**

__Formally__

**,**

**a**Markov

**partition**is used to provide

**a**

__finite__cover for the smooth system ; each

**set**

**of**the cover is associated with

**a**single symbol

**,**

**and**the sequences

**of**symbols result as

**a**trajectory

**of**the system moves from one

**of**the covering sets to another

**.**

__Formally__

**,**assuming our intuition from the

__finite__dimensional case carries over

**into**the infinite dimensional setting

**,**the measure should then be equal to

__Formally__

**,**the upper density

**of**

**a**graph G is the infimum

**of**the values α

**such**

**that**the

__finite__subgraphs

**of**G with density α have

**a**bounded number

**of**vertices

**.**

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__

**,**

**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

**.**

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