From 59053e9dfdb504055b652cd10ddfe4d01cc41051 Mon Sep 17 00:00:00 2001 From: Alexander Luzgarev Date: Sun, 16 Oct 2016 16:50:11 +0300 Subject: [PATCH] Add first colloquium questions --- colloq-1.pdf | Bin 0 -> 35234 bytes colloq-1.tex | 76 +++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 76 insertions(+) create mode 100644 colloq-1.pdf create mode 100644 colloq-1.tex diff --git a/colloq-1.pdf b/colloq-1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d0b3a5f5dfa4360e3530e571f3999168929e25bd GIT binary patch literal 35234 zcma%>Q*bT}45n+_w%b$Nwrzg3acbMPZQHhOyFInt{dZ?~_F``~lZ#|>oq3YHAy*U? z|HDMj217o-JhBeMLc~a9Z)^p_%L~IGYieiiVnM{j!NEoJzYK;!+|tIy)QN~e+{Vbo zRMgbO9$*T?&ky75;$&)M3*)|7qCI6t%K^8uS9iAvaTuYWJbGOy7~s0_*Hz9}{^(nwPv2WBFwEus&Hj^xZsJ`AI*y#O-TTYy{qqA` z?=3pMP2)X!<6v*YZxqG;73-)KWbjWiZB1lbz5a}sPpg!q+GH!w*fpPRUd|Q$U$yV# z+GmfQuAVha8-}=zV#xV3 zWp>A2Y~p$1jDF(1Q(7yi>n)w~Jk&ni8kKae7XoMf{;;W?p$EzGcmuoFCkHhqQg`x! 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Александр Лузгарев) +\par\medskip +Первый семестр, осень 2016 +\end{center} + +\glava{Наивная теория множеств} +\begin{compactenum} +\item Множества, подмножества, основные операции над множествами. +\item Отображения: образ, прообраз, инъекция, сюръекция, биекция. +\item Композиция отображений, ее ассоциативность, тождественное + отображение. +\item Левая/правая обратимость и инъективность/сюръективность. +\item График отображения, бинарные отношения и отношения эквивалентности. +\item Теорема о разбиении на классы эквивалентности. Фактор-множество. +\item Метод математической индукции. Бинарные операции. +\item Нейтральные элементы и обратимость. +\item Теорема об обобщенной ассоциативности. +\suspend{compactenum} +\glava{Элементарная теория чисел} +\resume{compactenum} +\item Делимость: определения и простейшие свойства. Ассоциированность. +\item Теорема о делении с остатком. +\item Наибольший общий делитель; его существование и +единственность. Линейное представление НОД. +\item Алгорифм Эвклида. +\item Свойства НОД. Взаимная простота, свойства взаимно простых чисел. +\item Линейные диофантовы уравнения. Полное описание множества решений +уравнения с двумя неизвестными. +\item НОД нескольких чисел и критерий разрешимости линейного диофантова +уравнения с несколькими неизвестными. +\item Простые числа, их свойства. +\item Основная теорема арифметики. +\item Каноническое разложение. Приложения: НОД, число делителей. +\item Сравнения по модулю. Свойства. +\item Классы вычетов, действия над ними. +\item Определение кольца. Кольцо классов вычетов. +\item Нулевое кольцо. Делители нуля, области целостности, поля. +\item Критерий обратимости элемента кольца классов вычетов. Когда + кольцо классов вычетов является полем? +\item Китайская теорема об остатках. +\item Теорема Вильсона. +\item Функция Эйлера. Переформулировка китайской теоремы об остатках в +терминах колец классов вычетов. +\item Мультипликативность функции Эйлера. Формула для функции Эйлера. +\item Теорема Эйлера и малая теорема Ферма. +\end{compactenum} + +\end{document} +