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Conversely, if a set of edges forms a forest, then by repeatedly removing leaves from this forest it can be shown by induction that the corresponding set of columns is independent.

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## Some Related Sentences

Conversely and if

__Conversely__

**,**

__if__statistics were uncovered which contradicted

**a**cherished theory

**,**

**the**sources were denounced as faulty

**.**

__Conversely__

**,**experimenters would consider as impressive such statements as

**the**following

**,**which

**,**

__if__they turned out to

**be**hits

**,**are so unusual as to

**be**really significant:

__Conversely__

**,**

__if__he gives

**a**heavy rating to his own reading

**,**and finds more accurate facts in

**it**than in

**the**others

**,**

**a**point

**is**chalked up for

**the**intrinsic

**,**objective meaningfulness

**of**

**this**type

**of**mediumistic material

**.**

__Conversely__

**,**

**a**subset R defines

**a**binary function

__if__and only

__if__

**,**for any x in X and y in Y

**,**there exists

**a**unique z in Z such

**that**( x

**,**y

**,**z ) belongs to R

**.**

__Conversely__

**,**

__if__

**a**Boolean ring A

**is**given

**,**we

**can**turn

**it**into

**a**Boolean algebra

**by**defining x ∨ y := x + y + ( x · y ) and x ∧ y := x · y

**.**

__Conversely__

**,**

__if__

**the**second theory

**is**correct

**,**

**then**

**the**Athenians were merely reacting to

**the**Persians attacking them

**.**

__Conversely__

**the**equilibrium position

**is**said to

**be**" far to

**the**left "

__if__hardly any product

**is**formed

**from**

**the**reactants

**.**

__Conversely__

**,**

__if__ƒ: C → C

**is**

**a**function which

**is**differentiable when regarded as

**a**function on R < sup > 2 </ sup >,

**then**ƒ

**is**complex differentiable

__if__and only

__if__

**the**Cauchy – Riemann equations hold

**.**

__Conversely__

**,**

__if__

**the**amount

**of**fat

**is**increased

**,**increasing

**the**number

**of**calories ingested will only add weight

**.**

__Conversely__

**,**

__if__

**the**source

**of**waves

**is**moving away

**from**

**the**observer

**,**each wave

**is**emitted

**from**

**a**position farther

**from**

**the**observer than

**the**previous wave

**,**so

**the**arrival time between successive waves

**is**increased

**,**reducing

**the**frequency

**.**

__Conversely__every partition

**of**comes

**from**an equivalence relation in

**this**way

**,**according to which

__if__and only

__if__and belong to

**the**same

**set**

**of**

**the**partition

**.**

If neither A nor B includes

**the**idea**of**existence**,****then**" some A are B " simply adjoins A to B**.**__Conversely__**,**__if__A or B do include**the**idea**of**existence in**the**way**that**" triangle " contains**the**idea " three angles equal to two right angles ",**then**" A exists "**is**automatically true**,**and we have an ontological proof**of**A's existence**.**__Conversely__

**,**

__if__

**the**measured potential

**is**negative

**,**

**the**electrode

**is**more oxidizing than

**the**SHE ( such as Zn in ZnSO < sub > 4 </ sub > where

**the**standard electrode potential

**is**− 0

**.**76 V ).

__Conversely__

**,**

__if__

**the**firm

**is**able to get bulk discounts

**of**an input

**,**

**then**

**it**could have economies

**of**scale in some range

**of**output levels even

__if__

**it**has decreasing returns in production in

**that**output range

**.**

__Conversely__

**,**daylight stock shot in daylight will look normal

**,**but orange

__if__shot under tungsten lights

**.**

__Conversely__

**,**

__if__

**the**data are sparse —

**that**

**is**

**,**

__if__only K out

**of**N Fourier coefficients are nonzero —

**then**

**the**complexity

**can**

**be**reduced to O ( K log N log ( N / K )), and

**this**has been demonstrated to lead to practical speedups compared to an ordinary FFT for N / K > 32 in

**a**large-N example ( N = 2 < sup > 22 </ sup >) using

**a**probabilistic approximate algorithm ( which estimates

**the**largest K coefficients to several decimal places ).

Conversely and set

__Conversely__

**,**any surjection between sets determines

**a**partition on its domain

**,**

**the**

__set__

**of**preimages

**of**singletons in

**the**codomain

**.**

__Conversely__

**,**given

**a**groupoid G in

**the**algebraic sense

**,**let G < sub > 0 </ sub >

**be**

**the**

__set__

**of**all elements

**of**

**the**form x * x < sup >− 1 </ sup > with x varying through G and define G ( x * x < sup >-1 </ sup >, y * y < sup >-1 </ sup >) as

**the**

__set__

**of**all elements f such

**that**y * y < sup >-1 </ sup > * f * x * x < sup >-1 </ sup > exists

**.**

__Conversely__

**,**

**the**symbol width could

**be**

__set__at 8

**,**even

**if**only values 0 and 1 are used ; these data would only require

**a**2-color table

**.**

__Conversely__

**,**one-dimensional orthogonal projectors are extreme points

**of**

**the**

__set__

**of**density operators

**.**

__Conversely__

**,**

**from**

**a**partial order on

**a**partition

**of**

**a**

__set__S one

**can**construct

**a**preorder on S

**.**There

**is**

**a**1-to-1 correspondence between preorders and pairs ( partition

**,**partial order ).

Each convex

__set__containing X must (**by****the**assumption**that****it****is**convex ) contain all convex combinations**of**points in X**,**so**the**__set__**of**all convex combinations**is**contained in**the**intersection**of**all convex sets containing X**.**__Conversely__**,****the**__set__**of**all convex combinations**is**itself**a**convex__set__containing X**,**so**it**also contains**the**intersection**of**all convex sets containing X**,**and therefore**the**sets given**by**these two definitions must**be**equal**.**
We define Thm ( C ) to

**be****the**__set__**of**all formulas**that**are valid in C**.**__Conversely__**,****if**X**is****a**__set__**of**formulas**,**let Mod ( X )**be****the**class**of**all frames which validate every formula**from**X**.**
It

**is****a**fact**that****the**ring**is****a**principal ideal ring ;**that****is****,**for any ideal I in**,**there exists an integer n in I such**that**every element**of**I**is****a**multiple**of**n**.**__Conversely__**,****the**__set__**of**all multiples**of**an arbitrary integer n**is**necessarily an ideal**,**and**is**usually denoted**by**( n ).
Given

**a**vector space V over**a**field K**,****the**span**of****a**__set__S ( not necessarily finite )**is**defined to**be****the**intersection W**of**all subspaces**of**V which contain S**.**W**is**referred to as**the**subspace spanned**by**S**,**or**by****the**vectors in S**.**__Conversely__**,**S**is**called**a**spanning__set__**of**W**.**__Conversely__

**,**given any ring

**,**we

**can**form

**a**category

**by**taking objects A < sub > n </ sub > indexed

**by**

**the**

__set__

**of**natural numbers ( including zero ) and letting

**the**hom-set

**of**morphisms

**from**to

**be**

**the**

__set__of-by-matrices over

**,**and where composition

**is**given

**by**matrix multiplication

**.**

__Conversely__

**,**

**a**phonetic character

__set__

**is**written precisely as

**it**

**is**spoken

**,**but with

**the**disadvantage

**that**slight pronunciation differences introduce ambiguities

**.**

__Conversely__

**,**once

**a**basis for A has been chosen

**,**

**the**products

**of**basis elements

**can**

**be**

__set__arbitrarily

**,**and

**then**extended in

**a**unique way to

**a**bilinear operator on A

**,**i

**.**e., so

**the**resulting multiplication satisfies

**the**algebra laws

**.**

__Conversely__

**,**

**if**R

**is**

**a**symmetric relation over

**a**

__set__X

**,**one

**can**interpret

**it**as describing an undirected graph with

**the**elements

**of**X as

**the**vertices and

**the**pairs in R as

**the**

**edges**

**.**

__Conversely__

**,**

**it**being

__set__apart implies

**the**opposite-that someone took

**the**clothes off in an ordinary manner ; furthermore

**,**

**the**Greek text uses

**the**word

**,**entetuligmenon

**,**" having been folded up ", seeming to imply some intentional action had been taken on

**the**soudarion

**.**

__Conversely__

**,**

**if**X

**is**

**a**Hausdorff space and ker f

**is**

**a**closed

__set__

**,**

**then**

**the**coimage

**of**f

**,**

**if**given

**the**quotient space topology

**,**must also

**be**

**a**Hausdorff space

**.**

__Conversely__

**,**techniques have been proposed for generating models

**of**GRNs

**that**best explain

**a**

__set__

**of**time series observations

**.**

Conversely and edges

__Conversely__

**,**given an ordered tree

**,**and conventionally draw

**the**root at

**the**top

**,**

**then**

**the**child nodes in an ordered tree

**can**

**be**drawn left-to-right

**,**yielding an essentially unique planar embedding ( up to embedded homotopy

**,**i

**.**e., moving

**the**

__edges__and nodes without crossing ).

To

**a**graph G there corresponds**a**signed complete graph Σ on**the**same vertex**set****,**whose__edges__are signed negative**if**in G and positive**if**not in G**.**__Conversely__**,**G**is****the**subgraph**of**Σ**that**consists**of**all vertices and all negative__edges__**.**

Conversely and forms

__Conversely__

**,**in

**the**English language as

**a**whole

**,**singular verb

__forms__

**can**often

**be**used with nouns ending in "- s "

**that**were once considered plural ( for example: " Physics

**is**my favorite academic subject ").

Given

**a**field ordering ≤ as in Def 1**,****the**elements such**that**x ≥ 0__forms__**a**positive cone**of**F**.**__Conversely__**,**given**a**positive cone P**of**F as in Def 2**,**one**can**associate**a**total ordering ≤< sub > P </ sub >**by**setting x ≤ y to mean y − x ∈ P**.**This total ordering ≤< sub > P </ sub > satisfies**the**properties**of**Def 1**.**__Conversely__

**,**nimbostratus

**is**coded as middle because

**it**usually initially

__forms__at mid-altitudes

**of**

**the**troposphere and becomes vertically developed

**by**growing downward into

**the**low altitude range

**.**

__Conversely__

**,**people not familiar with

**the**various different

__forms__

**of**dance may ask for

**a**evening

**of**square dance meaning simply

**a**barn dance where many different formations

**of**dance are used

**.**

__Conversely__

**,**there are exceptions to their ' original

__forms__being lost ' as in

**the**series

**,**Tobias was able to acquire his original DNA ; and hence was able to change into his original self

**.**

__Conversely__

**,**there have also been campaigns against potentially dangerous

__forms__

**of**contraception such as defective IUDs

**.**

*

__Conversely__**,**some authors might haphazardly switch between**the**subjunctive and indicative__forms__**of**verbs**,**with no intended difference in meaning**.**__Conversely__

**,**poets such as John Milton alternated between short and long

__forms__for function words

**,**depending on whether they occurred on or off

**the**meter

**.**

__Conversely__

**,**

**the**animated series adhered to

**the**character's intended transformation method

**of**splitting into two

__forms__

**,**save for an animation error in " Surprise Party ", which saw an unusually small Sky Lynx transform directly

**from**shuttle mode to lynx mode and in " The Quintesson Journal " when Sky Lynx transforms

**from**lynx mode to shuttle mode

**.**

__Conversely__

**,**she

**is**not as central in most

__forms__

**of**established Christianity as she

**is**in Gnosticism

**,**but to some Christian mystics

**the**Sophia

**is**

**a**very important concept

**.**

**is**

**a**volume form on M

**.**

__Conversely__

**,**given two volume

__forms__

**,**their ratio

**is**

**a**non-vanishing function ( positive

**if**they define

**the**same orientation

**,**negative

**if**they define opposite orientations ).

__Conversely__

**,**libertarians and anarcho-capitalists most commonly apply

**the**term " anarchism " to any philosophy

**that**opposes all

__forms__

**of**initiatory coercion ( without specifications regarding economic systems ).

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