From 9a1b3ba652e7e35898ce7830f596cfc91ee5bcb2 Mon Sep 17 00:00:00 2001 From: LinnAstapuk <13579pplena@gmail.com> Date: Thu, 26 Oct 2023 19:22:05 +0300 Subject: [PATCH] dmiml upd --- .vs/ProjectSettings.json | 3 + .vs/VSWorkspaceState.json | 6 + .vs/slnx.sqlite | Bin 0 -> 118784 bytes ...4745afa8-65ff-4183-a57f-9e934ee7e26a.vsidx | Bin 0 -> 538448 bytes ...de33e534-248b-497f-9b91-49deb1c0be28.vsidx | Bin 0 -> 34719 bytes .../FileContentIndex/read.lock | 0 .vs/summariesUSFAMCS/v17/.suo | Bin 0 -> 14336 bytes .../test.pdf" | Bin 0 -> 148639 bytes .../test.tex" | 218 +++ .../main.aux" | 26 + .../main.log" | 1494 +++++++++++++++++ .../main.out" | 2 + .../main.pdf" | Bin 271022 -> 304430 bytes .../main.synctex.gz" | Bin 0 -> 149516 bytes .../main.tex" | 115 ++ .../main.toc" | 3 + 16 files changed, 1867 insertions(+) create mode 100644 .vs/ProjectSettings.json create mode 100644 .vs/VSWorkspaceState.json create mode 100644 .vs/slnx.sqlite create mode 100644 .vs/summariesUSFAMCS/FileContentIndex/4745afa8-65ff-4183-a57f-9e934ee7e26a.vsidx create mode 100644 .vs/summariesUSFAMCS/FileContentIndex/de33e534-248b-497f-9b91-49deb1c0be28.vsidx create mode 100644 .vs/summariesUSFAMCS/FileContentIndex/read.lock create mode 100644 .vs/summariesUSFAMCS/v17/.suo create mode 100644 "\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/1 \321\201\320\265\320\274 \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207\320\260/test.pdf" create mode 100644 "\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/1 \321\201\320\265\320\274 \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207\320\260/test.tex" create mode 100644 "\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.aux" create mode 100644 "\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.log" create mode 100644 "\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.out" create mode 100644 "\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.synctex.gz" create mode 100644 "\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.toc" diff --git a/.vs/ProjectSettings.json b/.vs/ProjectSettings.json new file mode 100644 index 0000000..f8b4888 --- /dev/null +++ b/.vs/ProjectSettings.json @@ -0,0 +1,3 @@ +{ + "CurrentProjectSetting": null +} \ No newline at end of file diff --git a/.vs/VSWorkspaceState.json b/.vs/VSWorkspaceState.json new file mode 100644 index 0000000..6b61141 --- /dev/null +++ b/.vs/VSWorkspaceState.json @@ -0,0 +1,6 @@ +{ + "ExpandedNodes": [ + "" + ], + "PreviewInSolutionExplorer": false +} \ No newline at end of file diff --git a/.vs/slnx.sqlite b/.vs/slnx.sqlite new file mode 100644 index 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\320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207\320\260/test.tex" new file mode 100644 index 0000000..eac81b6 --- /dev/null +++ "b/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/1 \321\201\320\265\320\274 \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207\320\260/test.tex" @@ -0,0 +1,218 @@ +\documentclass[a4paper, 12pt]{report} +\usepackage[T2A]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage[english,russian]{babel} +\usepackage{amsmath,amsfonts,amssymb,amsthm,mathtools, tipa} +\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm,bindingoffset=0cm]{geometry} +\usepackage{ upgreek, mathrsfs, cancel} +\usepackage[unicode]{hyperref} +\usepackage{tikz} +\usepackage{colortbl} + +\newcommand{\RomanNumeralCaps}[1] + {\MakeUppercase{\romannumeral #1}} + +\newenvironment{Proof} % имя окружения для доказательства +{\par\noindent{$\blacklozenge$}} % символ рядом с \begin +{\hfill$\scriptstyle\boxtimes$} % символ рядом с \end + +\newenvironment{example} % имя окружения для примеров +{\par\noindent{\textsc{\textbf{Пример}}}} % символ рядом с \begin +%{\hfill$\scriptstyle\Box$} % символ рядом с \end + +\newenvironment{exercise} +{\par\noindent{\textsc{\textbf{Упражнение}.}}} +{\hfill} + +\newtheorem*{theorem}{Теорема} % окружение для теорем +\newtheorem*{corollary}{Следствия} % окружение для следствий +\newtheorem*{lemma}{Лемма} % окружение для лемм + +\newcommand{\Rm}{\mathbb{R}} +\newcommand{\Cm}{\mathbb{C}} +\newcommand{\I}{\mathbb{I}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\Q}{\mathbb{Q}} +% команды для множеств + + +\title{\textbf{\Huge{Дискретная математика и математическая логика}}\\Конспект по 1 семестру + специальностей «экономическая кибернетика» и «компьютерная безопасность»\\(лектор В. И. Бенедиктович)} % оформление титульного листа +\date{} % отключение даты в титульнике +\begin{document} + \maketitle % отображение титульного листа в документе + \tableofcontents{} +\chapter{Высказывания} % новая глава +\section*{Высказывания, операции над нами. Формулы логики высказываний (ФЛВ). Равносильные формулы, тавтологии, противоречия. Теорема о подстановке формулы вместо переменной. Теорема о замене подформулы равносильной ей формулой} +\subsection*{Высказывания} +$\bullet$ \textbf{Высказывание} - повествовательное предложение, относительно которого можно сделать вывод, что его содержание истинно или ложно (далее \textbf{И} - истинно, \textbf{Л} - ложно).\\\\ +Свойства высказываний: +\begin{enumerate} + \item \textbf{Закон исключения третьего}\\ + Всякое высказывание является либо истинным, либо ложным. + \item \textbf{Закон непротиворечивости}\\ + Никакое высказывание не может быть одновременно быть истинным и ложным. +\end{enumerate} + +\begin{example} + \begin{enumerate} + \item Сейчас дождь (Л); + \item $2 + 3 = 5$ (И); + \item $2 + 3 > 5$ (Л); + \item Закройте дверь (не высказывание); + \item Идёт ли дождь? (не высказывание); + \end{enumerate} +\end{example} +Обозначаем высказывания большими латинскими буквами ($A, B,\dotso, Z$). +\subsection*{Логические операции} +Из имеющихся высказываний можно получить другие спомощью логических операций\\ +\textbf{Логические операции}: +\begin{enumerate} + \item \textbf{Отрицание}\\ + $A$ - некоторое высказывание\\ + Высказывание типа: «неверно, что $A$» ($\bar A$/ $\neg A$)\\\\ + Таблица истинности:~~ + \begin{tabular}{|c|c|} + \hline + $A$ & $\overline{A}$ \\ + \hline + И & Л\\ + Л & И\\ + \hline + \end{tabular} + \item \textbf{Конъюнкция}\\ + Пусть $A$ и $B$ - некоторые высказывания\\ + Конъюнкцией высказываний $A$ и $B$ называется высказывание, которое обозначается $A \wedge B$ или $A \cdot B$ и которое принимает значение истинности тогда и только тогда, когда оба значения ($A$ и $B$) принимают значение «истинно».\\\\ + Таблица истинности:~~ + \begin{tabular}{|c|c|c|} + \hline + $A$ & $B$ & $A \wedge B$ \\ + \hline + И & И & И\\ + И & Л & Л\\ + Л & И & Л\\ + Л & Л & Л\\ + \hline + \end{tabular} + \item \textbf{Дизъюнкция}\\ + Пусть $A$ и $B$ - некоторые высказывания\\ + Дизъюнкцией этих высказываний, которое обозначается $A \vee B$, называется высказывание, которое принимает значение «истинно» тогда и только тогда, когда хотя бы одно из высказываний истинно\\\\ + Таблица истинности:~~ + \begin{tabular}{|c|c|c|} + \hline + $A$ & $B$ & $A \vee B$ \\ + \hline + И & И & И\\ + И & Л & И\\ + Л & И & И\\ + Л & Л & Л\\ + \hline + \end{tabular} + \item \textbf{Импликация}\\ + Пусть $A$ и $B$ - некоторые высказывания\\ + Импликация - высказывание, обозначается $A \Rightarrow B$, типа «если $A$, то $B$», которое принимает значение «ложь», когда высказывание $A$ - истинно, а $B$ - ложно.\\ + + Также $A$ - посылка, $B$ - заключение.\\\\ + Если импликация является истинной, то $B$ - необходимое условие для $A$, либо $A$ является достаточным условием для $B$ , либо $A$ влечёт $B$. + Если импликация является ложной, то из $A$ не следует $B$ ($A \nRightarrow B$)\\\\ + Таблица истинности:~~ + \begin{tabular}{|c|c|c|} + \hline + $A$ & $B$ & $A \Rightarrow B$ \\ + \hline + И & И & И\\ + И & Л & Л\\ + Л & И & И\\ + Л & Л & И\\ + \hline + \end{tabular} + \begin{center} + \textbf{Свойства импликации:} + \end{center} + \begin{enumerate} + \item транзитивность:\\ + $D = ((A \Rightarrow B) \wedge (B \Rightarrow C)) \Rightarrow (A \Rightarrow C)$ - принимает значение \textbf{И} при любых наборах $A, B, C$ + + + + \end{enumerate} + \item Эквивалентность + Высказывание $ A $ называют эквивалентным высказыванию $ B $ ,если выполняется + $ A $ необходимое и достаточное условие для $ B $.\\\\ + Таблица истинности:~~ +\begin{tabular}{|c|c|c|} + \hline + $A$ & $B$ & $A \Leftrightarrow B$ \\ + \hline + И & И & И\\ + И & Л & Л\\ + Л & И & Л\\ + Л & Л & И\\ + \hline +\end{tabular} + +\end{enumerate} + + +\begin{center} +Соглашение о приоритетах логических операций +\end{center} +1) Отприцание приоритетнее всех остальных операций\\ +2) Конъюнкция приоритетнее 3)\\ +3) Дизъюнкция приоритетнее 4-5)\\ +4) Импликация приоритетнее 5)\\ +5) Эквивалентность\\\\ +\begin{center} + \textbf{Понятие пропозициональной формулы } +\end{center} +- выражение, построенное из пропозициональных букв $ A, B, C, \dotso $ по следующим правилам: +\begin{enumerate} + \item Все буквы пропозициональны и является пропозициональной формулой +\item Если $ A, B, C $ – пропозициональные формулы, то и выражения с логическими операциями тоже являются пропозициональными формулами +\item Других формул нет +\end{enumerate} +обозначение пропозициональной формулы: \\ +Пропозициональная формула определяется на множестве всех возможных наборов значений переменных функции принимающие аргументы И Л\\ +Такая функция может быть задана с помощью конечной таблицы истинности, содержащей $2^n$ строк. +Формулы, выражающие одну и ту же формулу, принимают дно и то же значение, называют эквивалентными (одна и та же таблица истинности). + +Примеры эквивалентных формул:\\ +\begin{enumerate} + \item $ \neg (\neg X) = X$ (закон двойного отрицания); + \item $X\vee Y=Y\vee X$ (коммутативность дизъюнкции); + \item $X\wedge Y = Y\wedge X$ (коммутативность конъюнкции); + \item $(X\vee Y)\vee Z= X\vee(Y\vee Z)$ (ассоциативность дизъюнкции); + \item $(X\wedge Y) \wedge Z= X\wedge (Y\wedge Z)$ (ассоциативность конъюнкции); + \item $X\wedge (Y\vee Z)=(X \wedge Y)\vee(X\wedge Z)$ (дистрибутивность конъюнкции относительно дизъюнкции); + \item $X\vee(Y\wedge Z)=(X\vee Y)\wedge (X\vee Z)$ (дистрибутивность дизъюнкции относительно конъюнкции); + \item $X\vee X= X$ (закон идемпотентности дизъюнкции); + \item $X\wedge X = X$ (законы идемпотентности конъюнкции); + \item $X\vee \text{Л} = Ч$; + \item $X \wedge \text{Л} = \text{Л }$; + \item $\text{И} \wedge X=X$; + + + \item $X\vee \top = \top$; + \item $X\vee(\neg X)=\top$; + \item $X (\neg X)=\text{Л}$; + \item $\neg(X\wedge Y)=(\neg X)\vee(\neg Y)$; + \item $\neg(X\vee Y)=(\neg X) (\neg Y)$ (законы двойственности, или де Моргана); + \item $(X\Rightarrow Y)= (\neg X)\vee Y$; + \item $(X\Leftrightarrow Y)=(x\Rightarrow Y) (Y\Rightarrow X)= (\neg X \vee Y) \wedge (X\Rightarrow Y)$; + \item $(X\Rightarrow Y)= (\neg Y\Rightarrow \neg X)$ (закон обращения, или контрапозиции). + \item $X\vee (X\wedge Y)= X$ (закон поглощения относительно дизъюнкции); + \item $X (X\vee Y) = X$ (закон поглощения относительно конъюнкции); + \item $X\Rightarrow (Y \Rightarrow Z) = (X\wedge Y) \Rightarrow Z$ (закон объединения посылок); + \item $X\Rightarrow (Y \Rightarrow Z) = Y \Rightarrow (X\Rightarrow Z)$ (закон перестановки посылок); + \item $(A \wedge X)\vee (A \wedge \neg X)= $ (элементарное склеивание); + \item $(A \wedge X)\vee (B \wedge \neg X)=(A \wedge X)\vee (B \wedge (\neg X))\vee (A \wedge B)$ (обобщенное склеивание). + + + +\end{enumerate} + + + + +\end{document} diff --git "a/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. 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LaTeX Error: Unicode character ⊂ (U+2282) + not set up for use with LaTeX. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.576 ...раф} — граф $G'$, где $V'⊂ + V$ и/или $E⊂E'$.\\\\ +You may provide a definition with +\DeclareUnicodeCharacter + + +! LaTeX Error: Unicode character ⊂ (U+2282) + not set up for use with LaTeX. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.576 ...ф $G'$, где $V'⊂V$ и/или $E⊂ + E'$.\\\\ +You may provide a definition with +\DeclareUnicodeCharacter + + +! LaTeX Error: Unicode character ⊂ (U+2282) + not set up for use with LaTeX. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.577 Если $V'⊂ + V(G)$, то \textbf{порожденный (индуц... + +You may provide a definition with +\DeclareUnicodeCharacter + + +Underfull \hbox (badness 10000) in paragraph at lines 569--583 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 569--583 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 569--583 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 569--583 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 569--583 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 569--583 + + [] + + +Overfull \hbox (21.66182pt too wide) in paragraph at lines 569--583 +\T2A/cmr/m/n/12 $\OML/cmm/m/it/12 V[]V\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 G\ +OT1/cmr/m/n/12 )$\T2A/cmr/m/n/12 , \T2A/cmr/bx/n/12 --- (-- +---) - \T2A/cmr/m/n/12 -- - $\OML/cmm/m/it/12 V[] +G\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 V[]\OT1/cmr/m/n/12 ] = + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 569--583 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 569--583 + + [] + +[14] + +LaTeX Warning: Command \textendash invalid in math mode on input line 593. + + +Underfull \hbox (badness 10000) in paragraph at lines 584--596 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 584--596 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 584--596 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 601--605 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 606--607 + + [] + +[15] [16] (./main.aux) + +Package rerunfilecheck Warning: File `main.out' has changed. +(rerunfilecheck) Rerun to get outlines right +(rerunfilecheck) or use package `bookmark'. + +Package rerunfilecheck Info: Checksums for `main.out': +(rerunfilecheck) Before: +(rerunfilecheck) After: A267C854D9F78744DD554B243ECD89B9;302. + ) +Here is how much of TeX's memory you used: + 13835 strings out of 475071 + 213711 string characters out of 5782907 + 579513 words of memory out of 5000000 + 34960 multiletter control sequences out of 15000+600000 + 492236 words of font info for 85 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 72i,10n,77p,822b,620s stack positions out of 10000i,1000n,20000p,200000b,200000s +{c:/texlive/2022/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t2a.enc} +Output written on main.pdf (17 pages, 304430 bytes). +PDF statistics: + 236 PDF objects out of 1000 (max. 8388607) + 177 compressed objects within 2 object streams + 64 named destinations out of 1000 (max. 500000) + 1 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git "a/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. 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zz&wu(uazLBGP^etp~gg{ovNJNDqNf@i37KOL&yy_X;X)=r;;{xu)N$VAEVjJPjOZg zf(H&*e+(Do2$(ltgG)ohyPcAdvs7wV?yCfSE;}MEK{|pIla{DtFJ&QEC2En;8YjZ* zi7?HPVDd!xq6o6K8ui7ystZ{^D2v z`S1Vn@Bi!1?9cw`|NgK4{j>k{%b)$1U;gYr{=@(HhyVVM|MY+U=YRXh|M#n3{_3~C z{_EfU^4EX*_QTJs{o&95<_~}T^>6>}zyGVZAO7rb{`9*)wgig#ho8ILNI(0-zx(}P V|Lq_D^xL-|{y(t7*?Vac3IIT9+)4ld literal 0 HcmV?d00001 diff --git "a/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.tex" "b/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.tex" index 6aaeac9..a5fe542 100644 --- "a/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.tex" +++ "b/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.tex" @@ -499,5 +499,120 @@ $\bar x \bar y \vee y z \vee x \bar z$ и\\ $y \bar z \vee \bar x z \vee x y$ + + + + + + + +\chapter{Теория графов} % новая глава +\section*{Основные понятия} + +$\bullet$ \textbf{Граф} - следующая упорядоченная пара: $G = (V, E)$, где $V$ - непустое множество, состоящая из вершин графа, а $E \subseteq V^{(2)}$, где $V^{(2)}$ - все двухэлементные подмножества из $V$.\\\\ +$V^{(2)} = \{ \{v, w \} | v, w \in V \}$\\\\ + +$\bullet$ Граф \textbf{конечный}, если множество вершин конечно$|V| = n$. \\\\ +Если $|E|=n$, то граф $G$ обозначается $G(n,m)$ \\\\ +$n=|V|$ - порядок графа $G$.\\ +$m=|E|$ - размер графа $G$.\\ +Если $n=1$, то граф тривиальный. \\ +$\bullet$ Граф \textbf{простой}, если он не имеет петель - $\{v, w\}$ и не имеет кратных ребер, то есть несколько пар $\{v, w\}$.\\\\ +Вершины $v$ и $w$ \textbf{смежные}, обозначают $v\tilde w$, если ребро $\{v, w\} \in E$.\\\\ +Ребро $e=\{v, w\}=vw$ инцидентно вершинам $v$ и $w$, а эти вершины $v,~w$ называют концами $e$.\\\\ +Два ребра называют смежными, если существует $ v \in V $, которой они инцидентны или существует их общий конец.\\\\ +Пусть есть $v \in V(G)$. тогда множество всех вершин $u$ таких, что $u \tilde v, ~~ N_G(v)= \{u \in V(G)|u\tilde v\}$ называют окружением вершины $v$.\\\\ +$N_G(v) \cup \{ v \} = N_G [ v ] = N[v]$ — замкнутое окружение вершины $v$. \\\\ + +$\bullet$ \textbf{Степень вершины} $ v ~~~~ deg_G(v) = |N_G(v)|$ равна числу рёбер, выходящих из данной вершины.\\\\ + +Если $V' \subseteq V(G)$, то окружение множества вершин $V'$~ - множество $N_G(V') = \underset{v \in V'}{\cup} N_G(v)$\\\\ + +$\bullet$ Вершина, которая смежна с любой вершиной из $V$ называется \textbf{доминирующей}, то есть $v \in V;~ \forall u \in V \setminus \{v\}, ~ v\sim u $\\\\ + +Если $v \in V$, такая что для любой $u \in V \setminus \{v\}$ $v \sim u$, то $v$ - доминирующая. + +$ \bullet $ \textbf{Доминирующее множество} ~ - множество $U$, такое что для любого $v \in V(G) \setminus \{u\}$ существует $u \in U$, такое что $v \sim u$. + +$ \bullet $ \textbf{Число доминирования} $\gamma(G)$~- мощность наименьшего доминирующего множества.\\ + +$ \bullet $ Если $\deg(v) = 0$, то $v$~-\textbf{ изолированная}.\\ + +$ \bullet $ Если $\deg(v) = 1$, то $v$~- \textbf{висячая} или \textbf{лист}.\\ + +Минимальная степень вершин~ - $\delta(G) \geq 0$.\\ + +Максимальная степень вершин~ - $\Delta(G) \leq n - 1$.\\ + +$ \bullet $ Список степеней, упорядоченный по возрастанию ~- \textbf{\textbf{степенная последовательность}} $\delta = d_1 \leq d_2 \leq \dots \leq d_n \leq \Delta$. + +\textbf{Регулярный/однородный граф} степени $k$~($k$-регулярный)~- граф, такой что $\deg(v) = k$, $v \in V$. При $k = 3$ граф кубический.\\\\ + +$ \bullet $ \textbf{Псевдограф} — граф, который может содержать петли и кратные ребра, кратные петли.\\\\ +$ \bullet $ Мультиграф — граф, который может содержать кратные ребра.\\\\ +$ \bullet $ Если $E⊂V^2$ (составлено из упорядоченных двоек/пар), то граф \textbf{G—ориентированный} (орграф), а его ребра — дуги.\\\\ +Если $ v $ — начало дуги $(v,w)$, то дуга исходит из $v$.\\ +Если $ w $ — конец дуги $(v,w)$, то дуга заходит в $ w $.\\\\ + +$ \bullet $ Количество заходящих дуг — \textbf{полустепень захода} $ deg - w $. \\ + +$ \bullet $ Количество исходящих дуг — \textbf{полустепень исхода} $ deg + v $.\\ +Если в $ E $ существует состоящие из более чем двух вершин $e\{ v,~ w,~ u\}$, то $G$ —\textbf{гиперграф}.\\\\ +Для любого(мульти/псевдо) графа справедлива лемма: +\begin{lemma} Лемма о рукопожатиях:\\ +$ deg(v_1) + deg(v_2)+ \dotso + deg(v_n) = 2m, ~~~ \forall i,~~ v_i\in V. $ +\end{lemma} +\begin{Proof} +Это следует из того, что вклад каждого ребра в левую часть такой же, что и петли в правую часть равенства, равный двум. +\end{Proof} + +\textbf{Следствие}:\\ + Количество вершин нечетных степеней четно.\\\\ +$On$ — пустой граф — граф, состоящий из $ n $ изолированных вершин (нет ребер).\\\\ +$ Kn $ — полный граф — граф, где все вершины попарно смежны: $m=C_n^2$.\\\\ +$ Pn $ —цепь на $ n $ вершинах — граф, у которого $2$ листа, а остальные вершины имеют степень $2$. Длина (см. далее) равна $n-1$.\\\\ +$ Cn $ —цикл на $ n $ вершинах — связный (см. далее) граф, у которого все вершины имеют степень $ 2 $. +$G'=(V',E')$ — подграф графа $G$, если $V'\subseteq V,~~ E\subseteq E'$.\\\\ +$ \bullet $ \textbf{Собственный подграф} — граф $G'$, где $V'⊂V$ и/или $E⊂E'$.\\\\ +Если $V'⊂V(G)$, то \textbf{порожденный (индуцированный) подграф} множества вершин $V' G[V'] =def(V',E')$, где $E'={vu \in E(G) | v, u \in V'}$.\\ +$W\subseteq V(G) \Rightarrow G[V\setminus W]=def G-W$\\ +$F\subseteq E(G) \Rightarrow G+F = (V, E(G)\cup F) \Rightarrow G-F = (V, E(G)\setminus F)$\\ +$V=\{v\} \Rightarrow G - \{v\}$ аналогично $G - v$\\ +$F=\{e\} \Rightarrow G+\{e\}$ аналогично $G + e$\\ +$\Rightarrow G-\{e\}$ аналогично $G - e$\\\\ + +Независимое множество вершин $W\subseteq V(G)$ — $W$, такое что $G[W]$ - пустой. \\ +Мощность максимального $ W $ — число независимостей $ \alpha(G) $.\\ +Клика $W\subseteq V(G)$ — $W$, такое что $G[W]$ - полный.\\ +Мощность максимального $W$ — кликовое число $w(G)$.\\ +$ \bullet $ \textbf{Остов (субграф)} — подграф $G'=(V',E')$, такой что $V'=V(G)$.\\\\ +Пусть $G=(V,E)$ — произвольный подграф.\\ +Реберный граф — $L(G)$, такой что:\\ +1) вершины $L$ — ребра $G$\\ +2) $2$ вершины $ ij $ и $ kl $ принадлежащие $ E(G) $ — смежные, если в $ G~~ ij $ и $ kl $ — смежные.\\ +$ deg(ij) $ в $ L(G)=deg(i) $ в $G + deg(j) $ в $ G – 2 $.\\ +\textbf{Замечание}: порядок $ |L(G)|=m$, если $G(n,m) $.\\\\ +\textbf{Утверждение}: размер $ L(G)=m_L=\frac12 \cdot \sum\limits_{i=1}^n d_i^2 - m $.\\ +\begin{Proof} + Вершина $ i $ в $ G $ имеет степень $ d_i $, то в $ L(G) $ она образует $ C_{d_i}^2 $ ребер, каждая пара ребер—вершина в$ L(G) \Rightarrow m_L=\sum\limits_{i=1}^n C_{d_i}^2 = \frac12 \cdot \sum\limits_{i=1}^n d_i (d_i - 1) $=[по дистрибутивности и лемме о рукопожатиях] =$\frac12 \cdot \sum\limits_{i=1}^n d_i^2 - m $. +\end{Proof} + +\textbf{Следствие}:$ \sum\limits_{i=1}^n d_i^2 = \sum \limits_{ij\in E(G)} (d_i+d_j ) $ — первый индекс Загреба +\begin{Proof} +$ 2m_L=\sum \limits_{ij\in E(G)} d_ij =\sum \limits_{ij\in E(G)} (d_i + d_j - 2) =\sum \limits_{ij\in E(G)} (d_i + d_j ) - 2m $;\\ +$ 2(m_L+m)= \sum \limits_{ij\in E(G)} (d_i+d_j ) $ и $ 2(m_L+m)= \sum \limits_{i=1}^2 d_i^2 $ . +\end{Proof}\\ + +$ \bullet $ \textbf{Помеченный граф} — 1) граф, вершинам или ребрам которого присвоены какие-либо метки (числа, буквы); 2)граф порядка $ n $, если его вершинам присвоить попарно различные номера от $ 1 $ до $ n $.\\ +\begin{theorem} $1$\\ + Количество помеченных графов порядка $ n = 2^{C_n^2 } $. +\end{theorem} +\begin{Proof} + По определению количество ребер в полном графе порядка $ n $ равно числу всевозможных пар вершин равное $ C_n^2 \Rightarrow $ количество всех графов с фиксированным множеством вершин равное числу всех подмножеств множества всевозможных пар вершин = $ 2^{C_n^2} $. +\end{Proof} + + + + \end{document} \ No newline at end of file diff --git "a/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.toc" "b/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.toc" new file mode 100644 index 0000000..72c9396 --- /dev/null +++ "b/\320\224\320\234\320\270\320\234\320\233 1 \320\272\321\203\321\200\321\201 \320\222. \320\230. \320\221\320\265\320\275\320\265\320\264\320\270\320\272\321\202\320\276\320\262\320\270\321\207/main.toc" @@ -0,0 +1,3 @@ +\babel@toc {russian}{}\relax +\contentsline {chapter}{\numberline {1}Булевы функции}{2}{chapter.1}% +\contentsline {chapter}{\numberline {2}Теория графов}{13}{chapter.2}%

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