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\begin{document}
\title{Ba\u g\i nt\i lar}
\author{\"Oklid Geometrisine Giri\c s}
\date{5 Kas\i m 2018}
%\publishers{Matematik B\"ol\"um\"u\\
%Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
%\url{mat.msgsu.edu.tr/~dpierce/}\\
%\url{david.pierce@msgsu.edu.tr}}

\maketitle

%\tableofcontents

\section{Analiz}

Ger\c cel say\i lar i\c cin, i\c sareti $>$ olan
\textbf{daha b\"uy\"uk olma} ba\u g\i nt\i s\i\ s\i k s\i k kullan\i l\i yor.
\.Isimli iki \"ozelli\u gi vard\i r.
Daha b\"uy\"uk olma ba\u g\i nt\i s\i:
\begin{compactitem}
\item
  \textbf{ge\c ci\c slidir,}
  \c c\"unk\"u $a$, $b$, ve $c$ ger\c cel say\i\ olmak \"uzere
  \begin{equation*}
    a>b\And b>c\ise a>c;
  \end{equation*}
\item
  \textbf{yans\i mas\i zd\i r,}
  \c c\"unk\"u $a$ ger\c cel say\i\ olmak \"uzere
  \begin{equation*}
a>a\text{ de\u gildir.}
  \end{equation*}
\end{compactitem}
K\i saca ger\c cel say\i larda
\begin{align*}
  \Forall x\Forall y\Forall z&(x>y\land y>z\lto x>z),&
  \Forall x &x\not>x.
\end{align*}
\.I\c sareti $<$ olan
\textbf{daha k\"u\c c\"uk olma} ba\u g\i nt\i s\i n\i n
ayn\i\ \"ozelli\u gi vard\i r:
\begin{align*}
  \Forall x\Forall y\Forall z&(x<y\land y<z\lto x<z),&
  \Forall x &x\not<x.
\end{align*}
Bu iki ba\u g\i nt\i n\i n her biri,
di\u gerinin \textbf{tersidir:}
\begin{equation*}
  \Forall x\Forall y(x<y\liff y>x).
\end{equation*}
Tersi ayn\i\ olan bir ba\u g\i nt\i, \textbf{simetriktir.}
\"Orne\u gin e\c sitlik, simetriktir.

\.Iki ba\u g\i nt\i\ daha t\"uretilir:
\begin{equation*}
  \Forall x\Forall y\bigl(
  (x\geq y\liff x>y\lor x=y)\land(x\leq y\liff x<y\lor x=y)\bigr).
\end{equation*}
Daha b\"uy\"uk veya e\c sit olma ba\u g\i nt\i s\i:
\begin{compactitem}
\item
  ge\c ci\c slidir;
  \item
\textbf{yans\i mal\i d\i r,} \c c\"unk\"u $\Forall xx\geq x$;
\item
\textbf{ters simetrilidir,} \c c\"unk\"u
\begin{equation*}
  \Forall x\Forall y(x\geq y\land y\geq x\lto x=y).
\end{equation*}
\end{compactitem}
Bundan dolay\i,
daha b\"uy\"uk veya e\c sit olma,
bir \textbf{s\i ralamad\i r.}
Bu s\i ralama \textbf{do\u grusald\i r,}
\c c\"unk\"u
\begin{equation*}
  \Forall x\Forall y(x\geq y\lor y\geq x).
\end{equation*}
Ge\c ci\c sli ve yans\i mas\i z oldu\u gundan daha b\"uy\"uk olma,
bir \textbf{kesin s\i ralamad\i r.}
Ayr\i ca kesin \textbf{do\u grusal} bir s\i ralamad\i r,
\c c\"unk\"u
\begin{equation*}
  \Forall x\Forall y(x>y\lor x=y\lor y>x).
\end{equation*}


\section{Say\i lar Kuram\i}

E\u ger tamsay\i larda $a$ kere $b$, $c$ ederse, o zaman\\
\mbox{}\hfill
  $a$, $c$'yi \textbf{b\"oler;}
\hfill
  $b$, $c$'yi \textbf{\"ol\c cer.}
\hfill\mbox{}\\
\"Oklid'in
%\emph{\"O\u geler}'inin
\textsc{vii.}\ kitab\i n\i n 16.\ \"onermesinin sayesinde
\c carpma i\c slemi de\u gi\c smelidir,
dolay\i s\i yla b\"olme ba\u g\i nt\i s\i,
\"ol\c cme ba\u g\i nt\i s\i\ ile ayn\i d\i r.
Bu ba\u g\i nt\i n\i n i\c sareti $\divides$ oldu\u gundan
\begin{equation*}
  \Forall x\Forall y(x\divides y\liff\Exists zxz=y).
\end{equation*}
B\"olme ba\u g\i nt\i s\i, hem yans\i mal\i\ hem de ge\c ci\c slidir.

\emph{Sayma say\i lar\i nda} b\"olme, ters simetrilidir,
dolay\i s\i yla bir s\i ralamad\i r.
Bu s\i ralama do\u grusal de\u gildir,
\textbf{par\c cal\i d\i r.}

Tamsay\i larda,
Gauss'un 1801 y\i l\i nda verdi\u gi tan\i ma g\"ore,
e\u ger bir $a$ say\i s\i, $b$ ve $c$'nin fark\i n\i\ \"ol\c cerse,
o zaman $b$ ve $c$, \textbf{$a$ mod\"ul\"une g\"ore \c cak\i\c s\i r,}
ve
\begin{equation*}
  b\equiv c\pmod a
\end{equation*}
yaz\i l\i r.
Gauss'un Latince'sinde mod\"ul, \emph{modulus}'tur;
\emph{modulus,} k\"u\c c\"uk bir \emph{modus}'tur;
\emph{modus,} bir \"ol\c c\"ud\"ur.
K\i saca mod\"ul,
bir \"ol\c c\"uc\"ukt\"ur.
Bug\"un, verilen bir mod\"ule g\"ore \c cak\i\c s\i k say\i lara
\textbf{kalanda\c s} denebilir.

Her $n$ sayma say\i s\i\ i\c cin,
$n$ mod\"ul\"une g\"ore kalanda\c sl\i k ba\u g\i nt\i s\i\
\begin{inparaitem}
\item
  yans\i mal\i d\i r,
\item
  simetriktir,
\item
  ge\c ci\c slidir.
\end{inparaitem}
Sonu\c c olarak, tan\i ma g\"ore,
verilen bir mod\"ule g\"ore kalanda\c sl\i k,
bir \textbf{denklik ba\u g\i nt\i s\i d\i r.}

\"Oklid'in \emph{\"O\u geler}'inin
\textsc{vii.}\ kitab\i n\i n ba\c slang\i c\i nda bulunan
\emph{\"Oklid Algoritmas\i} ile
iki sayma say\i s\i n\i n
en b\"uy\"uk ortak b\"oleni
elde edilebilir.
\c Simdi
\begin{align*}
  \gcm(a,b)&=c,&\gcm(d,e)&=f
\end{align*}
olsun.
E\u ger
\begin{equation*}
  a=cx\land b=cy\land d=fx\land e=fy
\end{equation*}
sistemi \c c\"oz\"ulebilirse,
o zaman
\begin{equation*}
  (a,b)\sim(d,e)
\end{equation*}
yaz\i ls\i n.
Buradaki $\sim$ ba\u g\i nt\i s\i,
bir denklik ba\u g\i nt\i s\i d\i r.
Bir teoreme g\"ore
\begin{equation*}
  \Forall x\Forall y\Forall z\Forall w
  \bigl((x,y)\sim(z,w)\liff xw=yz\bigr).
\end{equation*}
Teoremi kan\i tlamak,
okura b\i rak\i l\i yor.

\section{Geometri}

Analizde ve say\i lar kuram\i nda
\c cok ifadeler, \emph{say\i} olarak d\"u\c s\"un\"ul\"ur.
\.Iki ifadenin say\i sal de\u geri ayn\i\ ise,
ifadeler \textbf{e\c sittir.}
Yukar\i da e\c sitli\u gin kavram\i n\i\ ve $=$ i\c saretini  zaten kulland\i k.
\"Orne\u gin ifade olarak $1+2$ ve $3$ farkl\i d\i r,
ama de\u geri ayn\i d\i r, dolay\i s\i yla
$1+2=3$.


\"Oklid geometrisinde a\c c\i lar
ve s\i n\i rlanm\i\c s do\u grular ve alanlar vard\i r,
ama hi\c cbirinin say\i sal de\u geri yoktur.
\"Oklid'in ``ortak kavramlar\i na'' g\"ore
\begin{compactitem}
\item
  birbiriyle \c cak\i\c san veya \"ort\"u\c sen,
  veya birbirine uygulayan, \c seyler e\c sittir;
\item
  e\c sitlik, bir denklik ba\u g\i nt\i s\i d\i r.
\end{compactitem}
\"Orne\u gin \"Onerme 4'e g\"ore
iki \"u\c cgende iki kenar, iki kenara e\c sit ise,
ve i\c cerilen a\c c\i lar da e\c sit ise,
o zaman \"u\c cgenler de e\c sittir,
\c c\"unk\"u \c cak\i\c s\i r.
\"Onerme 35'te \c cak\i\c smayan
ama \emph{par\c calar\i} \c cak\i\c san paralelkenarlar
e\c sittir.

\section{K\"umeler Kuram\i}

\.Iki k\"umenin elemanlar\i\ ayn\i\ ise,
k\"umeler \textbf{e\c sittir.}

Verilen bir $A$ k\"umesinde bir $B$ \textbf{ba\u g\i nt\i s\i,}
$A\times A$ veya $A^2$ \c carp\i m\i n\i n bir altk\"umesidir.
\"Orne\u gin $\R$ ger\c cel say\i lar k\"umesinde,
e\u ger pozitif say\i lar, $P$ k\"umesini olu\c sturursa,
o zaman
daha b\"uy\"uk olma ba\u g\i nt\i s\i,
\begin{equation*}
  \{(x,y)\in\R^2\colon x-y\in P\}
\end{equation*}
k\"umesidir.
Ayr\i ca $\Z$ tamsay\i lar k\"umesinde
bir $n$ mod\"ul\"une g\"ore kalanda\c sl\i k
\begin{equation*}
  \{(x,y)\in\Z^2\colon n\divides x-y\}
\end{equation*}
k\"umesidir.

Bir $A$ k\"umesinde $D$, %kalanda\c sl\i k veya e\c sitlik gibi
bir denklik ba\u g\i nt\i s\i\ olsun.
E\u ger $b\in A$ ise, o zaman tan\i ma g\"ore
$b$'nin \textbf{denklik s\i n\i f\i} veya \textbf{$D$ s\i n\i f\i,}
\begin{equation*}
  \{x\in A\colon x\mathrel Db\}
\end{equation*}
k\"umesidir.
Bu s\i n\i f, $[b]$ veya $\bar b$ veya $b/D$ olarak yaz\i labilir.

Bir \"ozel durumda, ba\c ska bir ifade vard\i r.
E\u ger $\sim$ ba\u g\i nt\i s\i\ yukar\i daki gibi tan\i mlan\i rsa,
o zaman $a$ ve $b$ sayma say\i s\i\ olmak \"uzere
$(a,b)$ s\i ral\i\ ikilisinin $\sim$ s\i n\i f\i,
$\frac ab$
veya $a/b$ \textbf{kesirli say\i s\i} olarak anla\c s\i labilir.

Genelde
\begin{equation*}
  [b]\cap[c]\neq\emptyset\lto  [b]=[c].
\end{equation*}
Bu sonuca bir kan\i t bulmak,
okura b\i rak\i l\i yor.



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