# Rustoda
Rustoda sotir ke ekavapafa solokseropa isu ovopa. Rustoda sotir leca volasa va megotunafa mukoda. Neken kreda zo roloplekud gu bata mukoda, tulon : belca ik dotay koegiskis vas ina. Va rustoda romalsavet ta kotcalara va yorda ko tazukoy ok ko drunot ok ta maera va miltay ik ar zvak. Rustoda rotir opelafa voda ok mukoda bro staroda ok nuok ok dace bro flioda. Koe jonoda ok tazuktaf enk ok dezotafa stoga, rustoda zo ikaplekur kan X megotunaca.
Va lan ord ke leca ta mukoda djuvolana gubeon favet ; tula : va eltay male i kale n ta bazexeem vox va eltay ke toleafu abavacku ta starodeem vox ε ta ageltota biwe kaike nedoy.

## Stravok ke rustoda vey gol
Koe opeem, gije zo sogesiar gu rustoda, bro gesiara va ugal gu {\displaystyle t} ok va tirka gu {\displaystyle x} maneka isu {\displaystyle y} isu {\displaystyle z}.
Ta patavara va abrotce is mante ke lavak dan krotc is ontine is  amiduca wale abrotce is mante zo gruped, va patavarinda ropimtat kan tula aze va yona tula rokawarzet ta kotafa pimtara va nuva. Bata nuva tiyir tela ikatcuna ta solokseropa koe Iraka vomi volalub-decem-bar-san-anyuste.
Ika origeem is benprostelanixeem, rogorat da va volgarifa voda ( rustoda ) ikaplekut gu yona leca. Numen rustoda sotir koeti tulaviawesi ko muxadjo dana zo rotikaplekur gu voda ; tula : gu ota.
Koe tula dragena gan solokseropa ke savsafa Iraka, ede K tir krotc isen o ontine isen a tir amiduca wale abrotce ( A ) is mante ( m ) pune
{\displaystyle A={\sqrt {\left({\frac {a}{2}}\right)^{2}+{\frac {K}{o}}}}+{\frac {a}{2}}\qquad \qquad m={\sqrt {\left({\frac {a}{2}}\right)^{2}+{\frac {K}{o}}}}-{\frac {a}{2}}}
Ede ikaplekut va a rustoda gu 6-e is va K gu 14-e is va o gu 2-e pune 
{\displaystyle A={\sqrt {\left({\frac {6}{2}}\right)^{2}+{\frac {14}{2}}}}+{\frac {6}{2}}\qquad \qquad m={\sqrt {\left({\frac {6}{2}}\right)^{2}+{\frac {14}{2}}}}-{\frac {6}{2}}}
trabe A=7 ( abrotce tir vas peroy ) isen m=1 ( mante tir vas tanoy ).

## Rustoda ke flioda
Digit va E belca isu F. Digit va {\displaystyle f} flioda gotuna cwade :
{\displaystyle {\begin{matrix}f:&E&\longrightarrow &F\\&x&\longmapsto &f(x)\end{matrix}}.}
{\displaystyle x} zo yoltar gu rustoda ke {\displaystyle f}({\displaystyle x}) muxaks.

### Tula
- Ta {\displaystyle f} flioda gotuna cwade :

{\displaystyle {\begin{matrix}f:&\mathbb {R} &\longrightarrow &\mathbb {R} \\&x&\longmapsto &3x+2\end{matrix}}}
{\displaystyle x} zo yoltar gu rustoda ke {\displaystyle f}({\displaystyle x}).
- Digit va {\displaystyle x=(x_{1};...;x_{n})\in \mathbb {R} ^{n}} . Ta {\displaystyle g} flioda gotuna cwade :

{\displaystyle {\begin{matrix}g:&\mathbb {R} ^{n}&\longrightarrow &\mathbb {R} \\&x&\longmapsto &{\sqrt {\sum _{i=1}^{n}x_{i}^{2}}}\end{matrix}}}
{\displaystyle x} tir rustoda ke {\displaystyle g}({\displaystyle x}). Dere rokalit da kota {\displaystyle x_{i}} ponasaca ke {\displaystyle x} tir rustoda ke {\displaystyle g}({\displaystyle x}). Kare vil, ont {\displaystyle g}({\displaystyle x}) va {\displaystyle x} rustoda acke tisa ton {\displaystyle n} lum digir, ont {\displaystyle g} tir flioda ke {\displaystyle n} rustoda dem lum vas 1.

## Nuyafa rustoda isu gluyanafa
Kalit da rustoda tir :
- nuyafa ede zo rotikaplekur kan yolt ke muka ke lan eem ; acum koe « 4x2 + x - 3 = 0 » fenkunaf tazukoy, xe tir nuyafa rustoda ; ede x zo ikaplekur kan a gronoda pune « 4a2 + a - 3 = 0 » muxaks tir budenaf slemaks ;
- gluyanafa viele va rapta ke sopuxa kofir, inde inafa yorda anton tir vas pimtara. Batinde tir icde x is k is i is t kottanon koe drageks bat :

{\displaystyle \forall x\in \mathbb {N} \quad x+1>0\quad ;\quad \sum _{k=1}^{b}k={\frac {b(b+1)}{2}}\quad ;\quad \prod _{i=1}^{10}i=3628800\quad ;\quad \pi =\int _{0}^{\infty }{\frac {2}{1+t^{2}}}\,dt\,}.
Kalit da ∀ sopuxa isu ∑ isu ∏ isu ∫ va rustoda sogluyad : sina sotid gluyasafa sugda.

### Yona tula

#### Taneafa tula
Rustoda gluyayana gan ∀ rustodagluyasiki gu tokoda soremsuted va kotikuca ke pilkoda, trabe va daniaskira va bata pilkoda gan kota mukoda ke lan ind.
Tula : katcalat da
{\displaystyle (1+0)^{2}\geq 1}
{\displaystyle (1+1)^{2}\geq 1+2\times 1}
{\displaystyle (1+(-0,5))^{2}\geq 1+2\times (-0,5)}
Nume va ropa roskut cwade :
ta kota ota {\displaystyle x\in \mathbb {R} },  {\displaystyle (1+x)^{2}\geq 1+2\times x}
Ede vey ovara bat ruyeks zo danedir pune va in di romalsavet ta beta narana ota. Va danedira va bat gekast, staper torigira va {\displaystyle x} rustoda volasa va konafa ageltota, is vonera va :
{\displaystyle (1+x)^{2}=1+2\times x+x^{2}\,}
Dere grupet da kota ageltota ekon gu tol- sotir dostaanaf numen  {\displaystyle x^{2}\geq 0}. Ison loplekuson vok bata dokotaca va {\displaystyle 1+2\times x}, pune 
{\displaystyle 1+2\times x+x^{2}\geq 1+2\times x}
numen 
{\displaystyle (1+x)^{2}\geq 1+2\times x}.
Numen pilkoda tir kotikafa.
Rustoda gluyayana gan ∃ rustodagluyasiki gu tokoda soremsuted va kruldera ke mukoda tuageltasa va lana pilkoda.
Tula : bat gekast :
« 2 melanibelcafa rontoda ke lana azenta va sint sogamdad koe 1 u »
soruyer da u tisu ke 2 melanibelcafa rontoda sokrulder, me tis kan tazukoy.
Ta danedira, vey 2 melanibelcafa rontoda va gekast rofavet aze roruyet da {\displaystyle M} u ke kottol sokrulder. Agelton {\displaystyle M} tir rustoda sovolasa va batu u isen gotuks va {\displaystyle M} rustoda di nover da vey batu u di kobat.

#### Toleafa tula
Digit va {\displaystyle f:\mathbb {R} \longrightarrow \mathbb {R} } is {\displaystyle x\in \mathbb {R} }. Bat valevef slemaks tagelton vas mila coba sosugdalad :
{\displaystyle i)\,\forall \epsilon >0,\exists \eta >0,\forall y\in \mathbb {R} ,|x-y|<\eta \Longrightarrow |f(x)-f(y)|<\epsilon }
{\displaystyle ii)\,\forall \epsilon >0,\exists \alpha >0,\forall y\in \mathbb {R} ,|x-y|<\alpha \Longrightarrow |f(x)-f(y)|<\epsilon }
{\displaystyle iii)\,\forall \epsilon >0,\exists \alpha >0,\forall \spadesuit \in \mathbb {R} ,|x-\spadesuit |<\alpha \Longrightarrow |f(x)-f(\spadesuit )|<\epsilon }
{\displaystyle iv)\,\forall y>0,\exists \epsilon >0,\forall \alpha \in \mathbb {R} ,|x-\alpha |<\epsilon \Longrightarrow |f(x)-f(\alpha )|<y}
{\displaystyle v)\,f\,flioda\,tir\,trenafa\,vas\,x}
Battode, {\displaystyle \epsilon ,\,\eta ,\,\alpha ,\,y,\,\spadesuit } rustoda tid gluyanafa. Batcoba en zo rokatcalanyar battode kiren slemaks en zo rovilder a favlara va sina.
Isen koe kota bata tula, {\displaystyle f} is {\displaystyle x} tid nuyafa rustoda. Keskeon, kota batcoba tir lidamafa gu :
Digit va {\displaystyle g:\mathbb {R} \longrightarrow \mathbb {R} } is {\displaystyle z\in \mathbb {R} }. Bat valevef slemaks tagelton vas mila coba sosugdalad :
{\displaystyle i)\,\forall \epsilon >0,\exists \eta >0,\forall y\in \mathbb {R} ,|z-y|<\eta \Longrightarrow |g(z)-g(y)|<\epsilon }
{\displaystyle ii)\,\forall \epsilon >0,\exists \alpha >0,\forall y\in \mathbb {R} ,|z-y|<\alpha \Longrightarrow |g(z)-g(y)|<\epsilon }
{\displaystyle iii)\,\forall \epsilon >0,\exists \alpha >0,\forall \spadesuit \in \mathbb {R} ,|z-\spadesuit |<\alpha \Longrightarrow |g(z)-g(\spadesuit )|<\epsilon }
{\displaystyle iv)\,\forall y>0,\exists \epsilon >0,\forall \alpha \in \mathbb {R} ,|z-\alpha |<\epsilon \Longrightarrow |g(z)-g(\alpha )|<y}
{\displaystyle v)\,g\,flioda\,tir\,trenafa\,vas\,z}
Isen ede tulon abduaykat va {\displaystyle f=\exp } is {\displaystyle x=0}, pune telyon daref slemaks vanpid drageks tis, battode, ageltaf.

## Solokseropafa rustoda isu givavuropafa
Koe dirgulafa avotxa ta talpeyara, coba dayoltana gan givavuropik gu rustoda tid vodafe tcalute towase darpe ugal. Numen batcoba kre tir pilkomara va runda ta setikera. Viele koe mil ind, va envaks va solokseropafa rustoda is envaks va givavuropafa gofaflat, bam tela givavuropafa rustoda zo dayoltar gu « runda » (« location » englavon). 
Koe fliodafa avotxa, tuke vuesterafa remawira, rustoda ke talpey sotid solokseropafa rustoda. 

## Izva
François Viète toz tutazuktayar faveson va eltay ta kaatoera va koeti favlanafi koe solokseropafa funtexa. Va xe jontikote favlat ta rustoda. Xanta nuve tir vey χ ellaseltay tis artazukaks va chay' (شيء) arabaravlem sosugdalas vas "coba".

## Rustodiskafa solokseropa
Moses Schönfinkel solokseropik diyir va rieta icde rozabdura va solokseropa vey rustodiskafa ovopa.  Ta batcoba, in va tazuktaf bolk dayoltan gu zinerafa ovopa al redur. Bat bolk al zo kanarir ise al zo tuvarar gan Haskell Curry solokseropik. Man bolk me dadir va esaca ke divedjafa ikabetara voxe ledrikon zo robelir. Favlason va patavara ke skeda, Tarski ovopik dere va rustodiskafa solokseropa al gotur.

Rustoda
Rustoda sotir ke ekavapafa [[solokseropa]] isu [[ovopa]]. Rustoda sotir leca volasa va megotunafa mukoda. Neken kreda zo roloplekud gu bata mukoda, tulon : [[belca]] ik dotay koegiskis vas ina. Va rustoda romalsavet ta kotcalara va yorda ko tazukoy ok ko [[drunot]] ok ta maera va miltay ik ar zvak. Rustoda rotir opelafa voda ok mukoda bro [[staroda]] ok [[Nouk|nuok]] ok dace bro [[flioda]]. Koe [[jonoda]] ok tazuktaf enk ok dezotafa stoga, rustoda zo ikaplekur kan ''X'' megotunaca.

Va lan ord ke leca ta mukoda djuvolana gubeon favet ; tula : va eltay male ''i'' kale ''n'' ta bazexeem vox va eltay ke toleafu abavacku ta starodeem vox ''ε'' ta ageltota biwe kaike nedoy.

== Stravok ke rustoda vey gol ==
Koe opeem, [[gije]] zo sogesiar gu rustoda, bro gesiara va ugal gu <math>t</math> ok va tirka gu <math>x</math> maneka isu <math>y</math> isu <math>z</math>.

Ta patavara va abrotce is mante ke lavak dan krotc is ontine is  amiduca wale abrotce is mante zo gruped, va patavarinda ropimtat kan tula aze va yona tula rokawarzet ta kotafa pimtara va nuva. Bata nuva tiyir tela ikatcuna ta solokseropa koe Iraka vomi volalub-decem-bar-san-anyuste.

Ika origeem is benprostelanixeem, rogorat da va volgarifa voda ( rustoda ) ikaplekut gu yona leca. Numen rustoda sotir koeti tulaviawesi ko muxadjo dana zo rotikaplekur gu voda ; tula : gu ota.

Koe tula dragena gan solokseropa ke savsafa Iraka, ede ''K'' tir krotc isen ''o'' ontine isen ''a'' tir amiduca wale abrotce ( ''A'' ) is mante ( ''m'' ) pune

:<math> A = \sqrt{\left(\frac{a}{2}\right)^2 + \frac{K}{o}} + \frac{a}{2} \qquad\qquad m = \sqrt{\left(\frac{a}{2}\right)^2 + \frac{K}{o}} - \frac{a}{2}</math>

Ede ikaplekut va ''a'' rustoda gu 6<sup>-e</sup> is va ''K'' gu 14<sup>-e</sup> is va ''o'' gu 2<sup>-e</sup> pune 
:<math> A = \sqrt{\left(\frac{6}{2}\right)^2 + \frac{14}{2}} + \frac{6}{2} \qquad\qquad m = \sqrt{\left(\frac{6}{2}\right)^2 + \frac{14}{2}} - \frac{6}{2} </math>

trabe ''A=7'' ( abrotce tir vas peroy ) isen ''m=1'' ( mante tir vas tanoy ).

== Rustoda ke flioda ==
Digit va ''E'' belca isu ''F''. Digit va <math>f</math> flioda gotuna cwade :

:<math> 
 \begin{matrix} f :&E & \longrightarrow & F 
\\& x & \longmapsto & f(x) \end{matrix}.</math>

<math>x</math> zo yoltar gu rustoda ke <math>f</math>(<math>x</math>) muxaks.

=== Tula ===
*Ta <math>f</math> flioda gotuna cwade :

:<math> \begin{matrix}f:& \R & \longrightarrow & \R 
\\ &x & \longmapsto & 3x+2 \end{matrix}
</math>

<math>x</math> zo yoltar gu rustoda ke <math>f</math>(<math>x</math>).


*Digit va <math>x=(x_1;...;x_n)\in\R^n</math> . Ta <math>g</math> flioda gotuna cwade :

<math> 
\begin{matrix} g : &\R^n & \longrightarrow & \R 
\\& x & \longmapsto & \sqrt{\sum_{i=1}^n x_i^2} \end{matrix}
</math>

<math>x</math> tir rustoda ke <math>g</math>(<math>x</math>). Dere rokalit da kota <math>x_i</math> ponasaca ke <math>x</math> tir rustoda ke <math>g</math>(<math>x</math>). Kare vil, ont <math>g</math>(<math>x</math>) va <math>x</math> rustoda acke tisa ton <math>n</math> lum digir, ont <math>g</math> tir flioda ke <math>n</math> rustoda dem lum vas 1.

== Nuyafa rustoda isu gluyanafa ==
Kalit da rustoda tir :

* ''nuyafa'' ede zo rotikaplekur kan yolt ke muka ke lan eem ; acum koe « 4''x''<sup>2</sup>&nbsp;+&nbsp;''x''&nbsp;-&nbsp;3&nbsp;=&nbsp;0 » fenkunaf tazukoy, xe tir nuyafa rustoda ; ede ''x'' zo ikaplekur kan '''a''' gronoda pune « 4'''a'''<sup>2</sup>&nbsp;+&nbsp;'''a'''&nbsp;-&nbsp;3&nbsp;=&nbsp;0 » muxaks tir budenaf slemaks ;
* ''gluyanafa'' viele va rapta ke sopuxa kofir, inde inafa yorda anton tir vas pimtara. Batinde tir icde ''x'' is ''k'' is ''i'' is ''t'' kottanon koe drageks bat :
<math>\forall x \in \mathbb N \quad  x+1> 0 \quad;\quad
\sum_{k=1}^b k  = \frac {b(b+1)}{2} \quad;\quad \prod_{i=1}^ {10} i = 3628800  \quad;\quad  \pi = \int_0^\infty \frac{2}{1+t^2} \,dt\, </math>.

Kalit da ∀ sopuxa isu ∑ isu ∏ isu '''∫''' va rustoda sogluyad : sina sotid '''gluyasafa''' sugda.

=== Yona tula ===

==== Taneafa tula ====
Rustoda gluyayana gan ∀ rustodagluyasiki gu tokoda soremsuted va kotikuca ke pilkoda, trabe va daniaskira va bata pilkoda gan kota mukoda ke lan ind.

Tula : katcalat da
:<math>(1+0)^2\geq 1</math>
:<math>(1+1)^2\geq 1+2\times 1</math>
:<math>(1+ (-0,5))^2\geq 1+2\times (-0,5)</math>
Nume va [[ropa]] roskut cwade :
:ta kota ota <math>x \in \mathbb R</math>,  <math>(1+x)^2\geq 1+2\times x</math>
Ede vey ovara bat ruyeks zo danedir pune va in di romalsavet ta beta narana ota. Va danedira va bat gekast, staper torigira va <math>x</math> rustoda volasa va konafa [[ageltota]], is vonera va :
:<math>(1+x)^2=1+2\times x+x^2\,</math>
Dere grupet da kota ageltota ekon gu tol- sotir dostaanaf numen  <math>x^2\geq 0</math>. Ison loplekuson vok bata dokotaca va <math>1+2\times x</math>, pune 
:<math>1+2\times x+x^2\geq 1+2\times x</math>
numen 
:<math>(1+x)^2\geq 1+2\times x</math>.
Numen pilkoda tir kotikafa.

Rustoda gluyayana gan ∃ rustodagluyasiki gu tokoda soremsuted va kruldera ke mukoda tuageltasa va lana pilkoda.

Tula : bat gekast :
:« ''2 melanibelcafa rontoda ke lana azenta va sint sogamdad koe 1 u ''»
soruyer da u tisu ke 2 melanibelcafa rontoda sokrulder, me tis kan tazukoy.

Ta danedira, vey 2 melanibelcafa rontoda va gekast rofavet aze roruyet da <math>M</math> u ke kottol sokrulder. Agelton <math>M</math> tir rustoda sovolasa va batu u isen gotuks va <math>M</math> rustoda di nover da vey batu u di kobat.

==== Toleafa tula ====
Digit va <math>f:\mathbb{R}\longrightarrow \mathbb{R} </math> is <math>x\in\mathbb{R}</math>. Bat valevef slemaks tagelton vas mila coba sosugdalad :

<math>i)\, \forall\epsilon>0,\exists\eta>0,\forall y\in\mathbb{R},|x-y|<\eta \Longrightarrow |f(x)-f(y)|<\epsilon</math>

<math>ii)\,\forall\epsilon>0,\exists\alpha>0,\forall y\in\mathbb{R},|x-y|<\alpha \Longrightarrow |f(x)-f(y)|<\epsilon</math>

<math>iii)\, \forall\epsilon>0,\exists\alpha>0,\forall \spadesuit\in\mathbb{R},|x-\spadesuit|<\alpha \Longrightarrow |f(x)-f(\spadesuit)|<\epsilon</math>

<math>iv)\,\forall y>0,\exists\epsilon>0,\forall \alpha\in\mathbb{R},|x-\alpha|<\epsilon \Longrightarrow |f(x)-f(\alpha)|<y</math>

<math>v)\,f\, flioda \, tir \, trenafa \, vas \, x</math>

Battode, <math>\epsilon,\,\eta,\, \alpha,\, y,\, \spadesuit</math> rustoda tid gluyanafa. Batcoba en zo rokatcalanyar battode kiren slemaks en zo rovilder a favlara va sina.

Isen koe kota bata tula, <math>f</math> is <math>x</math> tid nuyafa rustoda. Keskeon, kota batcoba tir lidamafa gu :

Digit va <math>g:\mathbb{R}\longrightarrow \mathbb{R} </math> is <math>z\in\mathbb{R}</math>. Bat valevef slemaks tagelton vas mila coba sosugdalad :

<math>i)\, \forall\epsilon>0,\exists\eta>0,\forall y\in\mathbb{R},|z-y|<\eta \Longrightarrow |g(z)-g(y)|<\epsilon</math>

<math>ii)\,\forall\epsilon>0,\exists\alpha>0,\forall y\in\mathbb{R},|z-y|<\alpha \Longrightarrow |g(z)-g(y)|<\epsilon</math>

<math>iii)\, \forall\epsilon>0,\exists\alpha>0,\forall \spadesuit\in\mathbb{R},|z-\spadesuit|<\alpha \Longrightarrow |g(z)-g(\spadesuit)|<\epsilon</math>

<math>iv)\,\forall y>0,\exists\epsilon>0,\forall \alpha\in\mathbb{R},|z-\alpha|<\epsilon \Longrightarrow |g(z)-g(\alpha)|<y</math>

<math>v)\,g\, flioda \, tir \, trenafa \, vas \, z</math>

Isen ede tulon abduaykat va <math>f=\exp</math> is <math>x=0</math>, pune telyon daref slemaks vanpid drageks tis, battode, ageltaf.

== Solokseropafa rustoda isu givavuropafa ==
Koe dirgulafa avotxa ta talpeyara, coba dayoltana gan givavuropik gu rustoda tid vodafe tcalute towase darpe ugal. Numen batcoba kre tir pilkomara va runda ta setikera. Viele koe mil ind, va envaks va solokseropafa rustoda is envaks va givavuropafa gofaflat, bam tela givavuropafa rustoda zo dayoltar gu « runda » (« ''location'' » englavon). 

Koe fliodafa avotxa, tuke vuesterafa remawira, rustoda ke talpey sotid solokseropafa rustoda. 

== Izva ==
[[François Viète]] toz tutazuktayar faveson va eltay ta kaatoera va koeti favlanafi koe solokseropafa funtexa. Va xe jontikote favlat ta rustoda. Xanta nuve tir vey χ ellaseltay tis artazukaks va ''chay''' (شيء) arabaravlem sosugdalas vas "coba".

== Rustodiskafa solokseropa ==

[[Moses Schönfinkel]] solokseropik diyir va rieta icde rozabdura va solokseropa vey rustodiskafa ovopa.  Ta batcoba, in va tazuktaf bolk dayoltan gu [[zinerafa ovopa]] al redur. Bat bolk al zo kanarir ise al zo tuvarar gan [[Haskell Curry]] solokseropik. Man bolk me dadir va esaca ke [[divedjafa ikabetara]] voxe ledrikon zo robelir. Favlason va patavara ke skeda, [[Alfred Tarski|Tarski]] ovopik dere va rustodiskafa solokseropa al gotur.

[[Loma:Solokseropa|Rustoda]]
[]