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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.

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## Some Related Sentences

Formally and define

__Formally__

**,**

**the**issue is that interfertile " able

**to**interbreed " is not

**a**transitive relation

**–**if A can breed with B

**,**

**and**B can breed with C

**,**it does not follow that A can breed with C

**–**

**and**thus does not

__define__

**an**equivalence relation

**.**

__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__however they

__define__it

**as**any variable that does not directly affect

**the**fundamentals

**of**

**the**economy

**.**

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 lines

__Formally__

**,**

**a**complex projective space is

**the**space

**of**complex

__lines__through

**the**origin

**of**

**an**

**(**n + 1 )- dimensional complex vector space

**.**

Formally and plane

__Formally__

**,**if is

**an**open subset

**of**

**the**complex

__plane__

**,**

**a**point

**of**

**,**

**and**is

**a**holomorphic function

**,**

**then**is called

**a**removable singularity for if there exists

**a**holomorphic function

**which**coincides with

**on**

**.**

__Formally__

**,**Aff

**(**V ) is naturally isomorphic

**to**

**a**subgroup

**of**

**,**with V embedded

**as**

**the**affine

__plane__

**,**namely

**the**stabilizer

**of**this affine

__plane__;

**the**above matrix formulation is

**the**

**(**transpose

**of**)

**the**realization

**of**this

**,**with

**the**

**(**n × n

**and**1 × 1 ) blocks corresponding

**to**

**the**direct sum decomposition

**.**

__Formally__

**,**

**a**frieze

**group**is

**a**class

**of**infinite discrete symmetry groups for patterns

**on**

**a**strip

**(**infinitely wide rectangle ), hence

**a**class

**of**groups

**of**isometries

**of**

**the**

__plane__

**,**or

**of**

**a**strip

**.**

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__

**,**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**partially ordered

**set**

**(**

__P__

**,**≤) is bounded complete if

**the**following holds for any subset S

**of**

__P__:

__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__

**,**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 L

__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__

**,**if there is

**a**utility function that describes preferences over

__L__commodities

**,**

**the**expenditure function

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