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Kombinatorika} +\end{center} + +\section{Permutáció} + +Elemek összes különböző sorbarendezéseinek a száma. Az elemek nem ismétlődhetnek. + +Sorrend $\Rightarrow$ IGEN + +Ismétléses $\Rightarrow$ IGEN + +Minden elem $\Rightarrow$ IGEN + +$$P_n = n!$$ + +A "képek" szó betűiből összeálítható összes szó: + +képek $\Rightarrow$ 5 betű (különbözők!) + +$P_5 = 5! = 120.$ + +\section{Permutáció Ismétléses} + +Elemek összes különböző sorbarendezéseinek a száma. Az elemek ismétlődhetnek! + +Sorrend $\Rightarrow$ IGEN + +Ismétléses $\Rightarrow$ IGEN + +Minden elem $\Rightarrow$ IGEN + + +$$P_n^{i_1, i_2, ... , i_r} = \frac{n!}{(i_1)!(i_2)! ... (i_r)!}$$ + +A "terep" szó betűiből összeálítható összes szó: + + +terep $\Rightarrow$ 5 betű (2 ugyan az!) + + +$P_5^{2} = \frac{5!}{2!} = \frac{5 * 4 * 3 * 2 * 1}{2 * 1} = 5 * 4 * 3 = 60$ + + +A "tollaslabda" szó betűiből összeálítható összes szó: + + +tollaslabda $\Rightarrow$ 11 betű (több ismétlődés van (3 * l, 3 * a) !) $\Rightarrow$ $P_{11}^{3, 3} = \frac{5!}{3! * 3!} = 1108800$ + + +\section{Variáció} + +Több elemből választunk kevesebb helyre. + +Sorrend $\Rightarrow$ IGEN + +Ismétléses $\Rightarrow$ IGEN + +Minden elem $\Rightarrow$ NEM + + +$$V_n^k = \frac{n!}{(n - k)!} = n * (n - 1) * (n - 2) * ... * (n - k + 1)$$ + +Rendszám, ha nem lehet ismétlődés: + +Betűk: $V_{26}^3 = \frac{26!}{26 - 3)!} = \frac{26!}{23!} = 26 * 25 * 24 = 15600$ + +Számok: $V_{10}^3 = \frac{10!}{10 - 3)!} = \frac{10!}{7!} = 10 * 9 * 8 = 720$ + +Az egész: $V_{26}^3 * V_{10}^3 = 11232000$ + +\section{Variáció Ismétléses} + +Több elemből választunk kevesebb helyre. + +Sorrend $\Rightarrow$ IGEN + +Ismétléses $\Rightarrow$ IGEN + +Minden elem $\Rightarrow$ NEM + + +$$V_n^{k, i} = n^k$$ + +Rendszám + +3 betű, 3 szám + +3 Betű: $V_n^{k, i} = V_{26}^{3, i} = 26^3 = 17576$ ($3 hely : 26 * 26 * 26$) +3 Szám: $V_n^{k, i} = V_{10}^{3, i} = 10^3 = 1000$ ($3 hely : 10 * 10 * 10$) + +$V_{26}^{3, i} * V_{10}^{3, i} = 26^3 * 10^3 = 17576000$ + +\section{Kombináció} + +Több elemből választunk kevesebb helyre. + +Sorrend $\Rightarrow$ NEM + +Ismétléses $\Rightarrow$ NEM + +Minden elem $\Rightarrow$ NEM + + +$$C_n^k = {{n}\choose{k}} = \frac{n!}{k!(n - k)!} $$ + +Pl szövegesen: + +A gyümölcssalátában van banán, alma, szőlő $\Rightarrow$ Kombináció + +3 Embert választunk 10 ből $\Rightarrow$ Kombináció + +Kiválasztunk egy zenekarba zongoristát, gitárost 10 emberből (a helyek miatt) $\Rightarrow$ Variáció + +Egy étterembe rendelünk 3 desszertet 10 ből $\Rightarrow$ Kombináció + +Felsorolni 3 külömböző kedvenc desszertet 10 ből $\Rightarrow$ Variáció + +A zár nyitó kombinációja 1233 volt $\Rightarrow$ Variáció + +Lottó $\Rightarrow$ Kombináció + + +\section{Kombináció Ismétléses} + +Több elemből választunk kevesebb helyre. + +Sorrend $\Rightarrow$ NEM + +Ismétléses $\Rightarrow$ IGEN + +Minden elem $\Rightarrow$ IGEN + + +$$C_n^{k, i} = C_{n + k -1}^k = {{n + k - 1}\choose{k}} = \frac{(n + k - 1)!}{k!((n + k - 1) - k)!} $$ + + +Ki akarunk választani 17 emberből egy hat tagú bizottságot, hány féleképpen lehet? + +$C_{17}^6 = C_{12 + 6 - 1}^6 = {{17 + 6 - 1}\choose{6}} = \frac{(17 + 6 - 1)!}{6!((17 + 6 - 1) - 6)!} = \frac{22!}{6!(22 - 6)!} = \frac{22!}{6!(22 - 6)!} = \frac{22!}{6! * 16!} = \frac{22 * 21 * 20 * 19 * 18 * 17}{6!} = 74613 $ + + + + + + +\end{document} \ No newline at end of file diff --git a/Diszkrét Matematika/Vizsga/VizsgaTetelek.tex b/Diszkrét Matematika/Vizsga/VizsgaTetelek.tex new file mode 100644 index 0000000..bea83d5 --- /dev/null +++ b/Diszkrét Matematika/Vizsga/VizsgaTetelek.tex @@ -0,0 +1,509 @@ +% Compile twice! + +\documentclass{beamer} +\usepackage{tikz} + +\usepackage[T1]{fontenc} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage[utf8]{inputenc} + +\usetheme{boxes} + + +\begin{document} + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Diszkrét Matematika}\\ + {\Large Vizsgatételek} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Halmazok, Relációk} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + + +\begin{frame} + +\begin{block}{Tétel: Minden dolog halmaza} +Nincs olyan halmaz, amelynek minden dolog eleme. +\end{block} + +\begin{block}{Biz} +asasdad +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Definíció: Unió} +Ha A és B halmazok, akkor A és B unióján a következő halmazt értjük:\\ +$$A \cup B = \{x | x \in A \vee x \in B\}$$ +\end{block} + +\begin{block}{Tétel: Az unió tulajdonságai} +Legyenek A, B, C tetszőleges halmazok. Ekkor: + +\begin{enumerate} +\item $A \cup \emptyset = A$ +\item $A \cup B = B \cup A$ (Kommutativitás) +\item $A \cup (B \cup C) = (A \cup B) \cup )$ (Asszociativitás) +\item $A \cup A = A$ (Idempotencia) +\item $A \subseteq B$ akkor, és csak akkor, ha $A \cup B = B$ +\end{enumerate} + +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Definíció: Metszet} +Ha A és B halmazok, akkor A és B metszetén a következő halmazt értjük:\\ +$$A \cap B = \{x \in A \wedge x \in B\}$$ +\end{block} + +\begin{block}{Tétel: A metszet tulajdonságai} +Legyenek A, B, C tetszőleges halmazok. Ekkor: + +\begin{enumerate} +\item $A \cap \emptyset = \emptyset$ +\item $A \cap B = B \cap A$ (Kommutativitás) +\item $A \cap (B \cap C) = (A \cap B) \cap C$ (Asszociativitás) +\item $A \cap A = A$ (Idempotencia) +\item $A \subseteq B$ akkor, és csak akkor, ha $A \cap B = A$ +\end{enumerate} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Unió és metszet disztributivitása} +Legyenek A, B, C tetszőleges halmazok. Ekkor: + +\begin{enumerate} +\item $A \cap (B \cup C) = (A \cap B) \cup (B \cap C)$ (A metszet disztributivitása az unióra nézve) + +\item $A \cup (B \cap C) = (A \cup B) \cap (B \cup C)$ (Az unió disztributivitása a metszetre nézve) +\end{enumerate} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Definíció: Komplementer} +Ha X halmaz, A $\wedge$ X, akkor A halmaz X-re vonatkoztatott komplementere:\\ +$$A' = X \setminus A$$ +\end{block} + +\begin{block}{Tétel: A komplementer tulajdonságai} +Legyenek A, B $\wedge$ X halmazok. Ekkor: + +\begin{enumerate} +\item $(A')' = A$ +\item $\emptyset' = X$ +\item $A \cap A' = \emptyset$ +\item $A \cup A' = X$ +\item $A \subseteq B$ akkor, és csak akkor, ha $B' \subseteq A'$ +\item $(A \cap B)' = A' \cup B'$ +\item $(A \cup B)' = A' \cap B'$ +\end{enumerate} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Definíció: Halmaz osztályfelbontása} +A tetszőleges X halmazt \textbf{osztályozzuk (osztályokra bontjuk)}, ha páronként diszjunkt, nemüres részhalmazainak uniójaként állítjuk elő. +\end{block} + +\begin{block}{Az X $\in$ X elem \textbf{ekvivalencia osztálya}:} +$$\overline{x} = \{y \in X : y \sim x\}$$ +\end{block} + +\begin{block}{Tétel: Ekvivalenciareláció és osztályfelbontás kapcsolata} +Valamely X halmazon értelmezett $\sim$ ekvivalenciareláció X-nek egy osztályfelbontását adja. Megfordítva, az X halmaz minden osztályfelbontása egy $\sim$ ekvivalenciarelációt hoz létre. +\end{block} + +\begin{block}{Biz} +asasdad +\end{block} + +\end{frame} + + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Algebrai struktúrák, számhalmazok} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Egységelem és inverz félcsoportban +Félcsoportban legfeljebb egy egységelem létezik, és minden elemnek legfeljebb egy, az egységelemre vonatkozó inverze létezik.} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Észrevételek gyűrűkben} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Nullosztó és regularitás} + +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Természetes számok} +Halmaz, egy nullér, és egy injektív unér művelettel (rákövetkezés) + + +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: N rendezése} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Felső határ és arkhimédészi tulajdonság} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Q nem felső határ tulajdonságú} +(12. dia lap alja) +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: $\sqrt{2}$ nem racionális} +Nincs Q-ban olyan szám, amelynek négyzete 2. +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + + +\begin{frame} + +\begin{block}{Tétel: Az algebra alaptétele} +\end{block} + +\end{frame} + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Számelmélet} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Az oszthatóság tulajdonságai EIT-ban} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Prím és irreducibilis elem EIT-ban} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Maradékos osztás Z-ben} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Prím és irreducibilis elem Z-ben} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: A számelmélet alaptétele} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Eukleidész tétele} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Kongruencia tulajdonságai} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Omnibusz tétel} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Euler-Fermat tétel} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: (Kis) Fermat tétel} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: A diofantikus egyenlet megoldása} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Kínai maradéktétel} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Számelméleti függvények} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: fi multiplikativitása} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: fi(n) kiszámolása} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Kombinatorika} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Véges halmaz valódi részhalmaza} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Skatulya-elv} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Permutációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Variációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Ismétléses variációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Kombinációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Ismétléses kombinációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Ismétléses permutációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Binomiális tétel} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Logikai szita formula} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + + + + + + + + +\begin{frame} + +\begin{block}{Tétel: Egységelem és inverz félcsoportban} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + + +\end{document} \ No newline at end of file diff --git a/Számítástudomány/Vizsga/szamtudv.tex b/Számítástudomány/Vizsga/szamtudv.tex new file mode 100644 index 0000000..8a0bc3a --- /dev/null +++ b/Számítástudomány/Vizsga/szamtudv.tex @@ -0,0 +1,509 @@ +% Compile twice! + +\documentclass{beamer} +\usepackage{tikz} + +\usepackage[T1]{fontenc} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage[utf8]{inputenc} + +\usetheme{boxes} + + +\begin{document} + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Diszkrét Matematika}\\ + {\Large Vizsgatételek} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Halmazok, Relációk} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + + +\begin{frame} + +\begin{block}{Tétel: Minden dolog halmaza} +Nincs olyan halmaz, amelynek minden dolog eleme. +\end{block} + +\begin{block}{Biz} +rrrrrrrrrrrrrrrrrr +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Definíció: Unió} +Ha A és B halmazok, akkor A és B unióján a következő halmazt értjük:\\ +$$A \cup B = \{x | x \in A \vee x \in B\}$$ +\end{block} + +\begin{block}{Tétel: Az unió tulajdonságai} +Legyenek A, B, C tetszőleges halmazok. Ekkor: + +\begin{enumerate} +\item $A \cup \emptyset = A$ +\item $A \cup B = B \cup A$ (Kommutativitás) +\item $A \cup (B \cup C) = (A \cup B) \cup )$ (Asszociativitás) +\item $A \cup A = A$ (Idempotencia) +\item $A \subseteq B$ akkor, és csak akkor, ha $A \cup B = B$ +\end{enumerate} + +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Definíció: Metszet} +Ha A és B halmazok, akkor A és B metszetén a következő halmazt értjük:\\ +$$A \cap B = \{x \in A \wedge x \in B\}$$ +\end{block} + +\begin{block}{Tétel: A metszet tulajdonságai} +Legyenek A, B, C tetszőleges halmazok. Ekkor: + +\begin{enumerate} +\item $A \cap \emptyset = \emptyset$ +\item $A \cap B = B \cap A$ (Kommutativitás) +\item $A \cap (B \cap C) = (A \cap B) \cap C$ (Asszociativitás) +\item $A \cap A = A$ (Idempotencia) +\item $A \subseteq B$ akkor, és csak akkor, ha $A \cap B = A$ +\end{enumerate} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Unió és metszet disztributivitása} +Legyenek A, B, C tetszőleges halmazok. Ekkor: + +\begin{enumerate} +\item $A \cap (B \cup C) = (A \cap B) \cup (B \cap C)$ (A metszet disztributivitása az unióra nézve) + +\item $A \cup (B \cap C) = (A \cup B) \cap (B \cup C)$ (Az unió disztributivitása a metszetre nézve) +\end{enumerate} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Definíció: Komplementer} +Ha X halmaz, A $\wedge$ X, akkor A halmaz X-re vonatkoztatott komplementere:\\ +$$A' = X \setminus A$$ +\end{block} + +\begin{block}{Tétel: A komplementer tulajdonságai} +Legyenek A, B $\wedge$ X halmazok. Ekkor: + +\begin{enumerate} +\item $(A')' = A$ +\item $\emptyset' = X$ +\item $A \cap A' = \emptyset$ +\item $A \cup A' = X$ +\item $A \subseteq B$ akkor, és csak akkor, ha $B' \subseteq A'$ +\item $(A \cap B)' = A' \cup B'$ +\item $(A \cup B)' = A' \cap B'$ +\end{enumerate} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Definíció: Halmaz osztályfelbontása} +A tetszőleges X halmazt \textbf{osztályozzuk (osztályokra bontjuk)}, ha páronként diszjunkt, nemüres részhalmazainak uniójaként állítjuk elő. +\end{block} + +\begin{block}{Az X $\in$ X elem \textbf{ekvivalencia osztálya}:} +$$\overline{x} = \{y \in X : y \sim x\}$$ +\end{block} + +\begin{block}{Tétel: Ekvivalenciareláció és osztályfelbontás kapcsolata} +Valamely X halmazon értelmezett $\sim$ ekvivalenciareláció X-nek egy osztályfelbontását adja. Megfordítva, az X halmaz minden osztályfelbontása egy $\sim$ ekvivalenciarelációt hoz létre. +\end{block} + +\begin{block}{Biz} +asasdad +\end{block} + +\end{frame} + + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Algebrai struktúrák, számhalmazok} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Egységelem és inverz félcsoportban +Félcsoportban legfeljebb egy egységelem létezik, és minden elemnek legfeljebb egy, az egységelemre vonatkozó inverze létezik.} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Észrevételek gyűrűkben} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Nullosztó és regularitás} + +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Természetes számok} +Halmaz, egy nullér, és egy injektív unér művelettel (rákövetkezés) + + +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: N rendezése} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Felső határ és arkhimédészi tulajdonság} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Q nem felső határ tulajdonságú} +(12. dia lap alja) +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: $\sqrt{2}$ nem racionális} +Nincs Q-ban olyan szám, amelynek négyzete 2. +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + + +\begin{frame} + +\begin{block}{Tétel: Az algebra alaptétele} +\end{block} + +\end{frame} + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Számelmélet} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Az oszthatóság tulajdonságai EIT-ban} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Prím és irreducibilis elem EIT-ban} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Maradékos osztás Z-ben} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Prím és irreducibilis elem Z-ben} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: A számelmélet alaptétele} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Eukleidész tétele} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Kongruencia tulajdonságai} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Omnibusz tétel} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Euler-Fermat tétel} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: (Kis) Fermat tétel} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: A diofantikus egyenlet megoldása} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Kínai maradéktétel} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Számelméleti függvények} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: fi multiplikativitása} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: fi(n) kiszámolása} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame}[plain] +\begin{tikzpicture}[overlay, remember picture] +\node[anchor=center] at (current page.center) { +\begin{beamercolorbox}[center]{title} + {\Huge Kombinatorika} +\end{beamercolorbox}}; +\end{tikzpicture} +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Véges halmaz valódi részhalmaza} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Skatulya-elv} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Permutációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Variációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Ismétléses variációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Kombinációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Ismétléses kombinációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Ismétléses permutációk száma} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Binomiális tétel} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + +\begin{frame} + +\begin{block}{Tétel: Logikai szita formula} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + + + + + + + + +\begin{frame} + +\begin{block}{Tétel: Egységelem és inverz félcsoportban} +\end{block} + +\begin{block}{Bizonyítás} +\end{block} + +\end{frame} + + +\end{document} \ No newline at end of file