From 8d53e6e409fcdc17633b271c203a652c713847d4 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Fri, 9 Jun 2023 10:48:40 +0200 Subject: [PATCH] Finished assignment 3, pretty pleased with the results actually, didn't think I would make it this far --- assignments/main_text.pdf | Bin 617785 -> 635483 bytes assignments/week3/main.pdf | Bin 487896 -> 492046 bytes assignments/week3/main.tex | 28 +++++++++++++++++++++++----- 3 files changed, 23 insertions(+), 5 deletions(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index 7011f7c05808f5eda7b5ce8709a90b5b3f50bcd0..0689e8b7f4cb6d7de939f475bd061a7b42ea1e04 100644 GIT binary patch delta 327581 zcmZs?Q*@wB(5@ZZwr$&**tRtjn@?=pwllHqiS1;PgmPXd#SXtN`QKvz%Sh+}8NE}RU;RFQWnB^_(tz4~1 zSlQEWSz+mb*x5!TFj5qVxxD%{G&mldHAvlLe)OMwW~Jqgg80Hlv0>IG&dVGQRJcmm zcg-4K$=PLLl`x@ubnF$mw83mvTI6c{^M>)yUaXSD^=<3g4Xg8O5y}PP9O11#SK1MB zLLax4@6D4g&)t)9&v$p9n=Be%LQwsfOy-nP=-sQtH)^=q-XNNiqQY<%_U10G&K4$i 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Yc6T%}g=gjDU}5Hj|M^o?P7MBk0N%P9MgRZ+ diff --git a/assignments/week3/main.tex b/assignments/week3/main.tex index 6e5ce1a..9398a03 100644 --- a/assignments/week3/main.tex +++ b/assignments/week3/main.tex @@ -74,7 +74,7 @@ $E$ to $\mathcal{P}(\N)$, $|E| = |\mathcal{P}(\N)|$. \qed discussed in the lectures that $\R\backslash\Q$ is infinite and $\R$ is uncountable without proof. \end{enumerate} \end{tcolorbox} -\begin{enumerate}[label=\emph{(\alph*)},wide] +\begin{enumerate}[label=\emph{(\alph*)}] \item So let $A$ and $B$ be two disjoint, countably infinite sets. Since these sets are countably infinite, a bijective function to $\N$ exists for both functions separately. It is then straightforward to map both these function together to $\Z$ instead, in the following way. Let $f$ be the bijective function such that @@ -152,7 +152,7 @@ Since the implication holds both ways, the equivalence is proven. \qed \end{enumerate} \end{tcolorbox} -\begin{enumerate}[label=\emph{(\alph*)},wide] +\begin{enumerate}[label=\emph{(\alph*)}] \item Since $\R$ is open, it is clear that $(-\infty, a)$ and $(b, \infty)$ are open to the left and right respectively as well. Also, their respective right and left side are present in $(a,b)$ as well, so we will only prove it for this case. The other cases follow logically. @@ -191,11 +191,29 @@ Since the implication holds both ways, the equivalence is proven. \qed \item Similarly to the previous exercise, non-formally speaking we want to prove that any intersection of open sets in $\R$ is open itself. This is not as trivial as in the previeous exercise however: since every set that is added as an intersection poses another restriction, we don't have the immediate guarantee that every $\epsilon$ - from the subsets will also hold for the intersection set. + from the subsets will also be a well-defined element for the intersection set. - Formally, let $n \in \N$ and let $U_1,...,U_n \subset \R$. We assume that all $U_1,...,U_n$ are open. + Now formally. Let $n \in \N$ and let $U_1,...,U_n \subset \R$. We assume that all $U_1,...,U_n$ are open. + Then we will prove that $\bigcap_{m = 1}^n U_m$ is open by induction over $n$. - \item + For the base case, let $n = 1$. Then the intersection set is equivalent to $U_1$. Since $U_1$ is open, then + so is the intersection set. + + For the inductive step, we assume that the intersection set is open for a certain $n = h$, i.e. there exists + an $\epsilon > 0$ such that $(x - \epsilon, x + \epsilon)$ is open for every $x \in \bigcap_{m = 1}^h U_m$. + Now, we will add one additional open set to this intersection, $U_{h+1}$, such that $n$ become $h+1$. Note + that $h + 1 \in \N$. Let the new intersection set be denoted as $\bigcap'$, and the old one as $\bigcap$. + Then in order to find an $\epsilon > 0$ for every $x \in \bigcap'$ such that $(x - \epsilon, x + \epsilon)$, + we take the smallest of $\epsilon$'s for that $x$ compared between $\bigcap$ and $U_{h+1}$. Since + $x \in \bigcap'$, we know that $x \in \bigcap$ and $x \in U_{h+1}$. Then the smallest accomponying $\epsilon$ + always gives a well-defined open set inside of $\bigcap'$ because $|(x + \epsilon_1) - (x - \epsilon_1)| < + |(x + \epsilon_2) - (x - \epsilon_2)|$ if $\epsilon_1 < \epsilon_2$, and thus $\bigcap'$ is open itself. \qed + \item No, $\Q$ is not open in $\R$. This is because we can't find an $\epsilon > 0$ such that for every $q \in \Q$, + $(q - \epsilon, q + \epsilon) \subset \Q$. We know that $\Q$ is dense in $\R$, but as we have proven in + exercise 1, the converse is also true. For every real numbers, we can find a real number inbetween that is not + a rational number. So, we cannot pick an $\epsilon > 0$ such that there is an interval around $x$ that itself + is completely contained in $\Q$. For every $\epsilon$ we pick, we can always find a real number $r$ such that + $x < r < x + \epsilon$ and $x - \epsilon < r < x$. So, $\Q$ is not open. \qed \end{enumerate} \exercise*

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