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+\begin{document} + +\section{Assignment 3} + +\exercise* + +\end{document} From 6fcd7eca69e0b7406d9d7562277f05b993af2356 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Fri, 26 May 2023 16:54:00 +0200 Subject: [PATCH 2/7] Finished first exercise of assignment 3 and did some refactoring on numbering etc --- assignments/main_text.pdf | Bin 602879 -> 607015 bytes assignments/main_text.tex | 1 - assignments/week1/main.pdf | Bin 487932 -> 487932 bytes assignments/week1/main.tex | 5 +++-- assignments/week2/main.pdf | Bin 469628 -> 471675 bytes assignments/week2/main.tex | 15 ++++++++------- assignments/week3/main.pdf | Bin 157320 -> 367776 bytes assignments/week3/main.tex | 28 +++++++++++++++++++++++++++- 8 files changed, 38 insertions(+), 11 deletions(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index 85fd2e93d83e27ec46a490ce40a986dd667990b2..ea9d43324fe2c890bbe3176554a58a0465f3d727 100644 GIT binary patch delta 87915 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Since $\N$ is countably infinite, Each set is definitely finite, because they all contain just one element. Finally, the union of the collection of sets is equal to $\N$, which is not a finite set. -\exercise[6] +\exercise* \begin{tcolorbox} \begin{enumerate}[label=\emph{\alph*)}, wide] \item Compute $f(4/15)$. 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index ad580a0..d5480de 100644 --- a/assignments/week2/main.tex +++ b/assignments/week2/main.tex @@ -1,7 +1,8 @@ \documentclass[../main_text.tex]{subfiles} \begin{document} +\setcounter{exercise}{0} -\section{Assignment 2} +\part{Assignment 2} \exercise*[1.1.1] \begin{tcolorbox} @@ -17,7 +18,7 @@ Working out the right side with the distributive law, gives $0 < (-x*z)-(-x*y)$. Using $-1*-1 = 1$, gives $0 < (-xz)-(-xy)$, thus $0 < xy - xz$. The right part can be split again: $xz < xy$. Then, the < can be flipped, which gives $xy > xz$. \qed -\exercise[1.1.2] +\exercise*[1.1.2] \begin{tcolorbox} Let $S$ be an ordered set. Let $A \subset S$ be a non-empty finite subset. Then A is bounded. Furthermore, $\text{inf} A$ exists and in in $A$ and $\sup A$ exists and is in $A$. @@ -45,7 +46,7 @@ of $A$ in $A$ \footnote{It might even be that I have already proven that $A$ has Then that's also good enough to show that $A$ is bounded, since in order for $A$ to have a supremum, it must also be bounded.}. This will be left to the reader. -\exercise[1.1.5] +\exercise*[1.1.5] \begin{tcolorbox} Let $S$ be an ordered set. Let $A \subset S$ and suppose $b$ is an upper bound for $A$. Suppose $b \in A$. Show that $b = \sup A$. @@ -63,7 +64,7 @@ not an upper bound of $A$ and thus also not its supremum. If $c > b$, then $b$ i $c$ and therefore $c$ cannot be the supremum. The only option left is that $c=b$, and therefore $b = \sup A$. \qed -\exercise[1.1.6] +\exercise*[1.1.6] \begin{tcolorbox} Let $S$ be an ordered set. Let $A \subset S$ be nonempty and bounded above. Suppose $\sup A$ exists and $\sup A \notin A$. Show that $A$ contains a countably infinite subset. @@ -79,7 +80,7 @@ $a < b$ and $x < a $ $\forall x \in A$. So $b$ is in fact not the least upper bo which is in contradiction with our assumption earlier. Ergo, $|X| \geq |\N|$. Since $X$ then is at least of the same cardinality as $\N$, it must also contain a countably infinite subset. \qed -\exercise[1.2.7] +\exercise*[1.2.7] \begin{tcolorbox} Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x, y$, we have \begin{equation*} @@ -103,7 +104,7 @@ Now to prove the second statement. We assume $x = y$ is a positive real number. \sqrt{xy}=\sqrt{x^2}=x=\frac{2x}{2}=\frac{x + y}{2}. \qed \end{equation*} -\exercise[1.2.9] +\exercise*[1.2.9] \begin{tcolorbox} Let $A$ and $B$ be two nonempty bounded sets of real numbers. Define the set $C := \{a + b : a \in A, b \in B\}$. Show that $C$ is a bounded set and that @@ -133,7 +134,7 @@ $y$ can be given by the supremum; $x \leq \sup A$ and $y \leq \sup B$. So, $\sup A similar argument can be given for the infimum, which is left to the reader. -\exercise[7] +\exercise* \begin{tcolorbox} Let \begin{equation*} diff --git a/assignments/week3/main.pdf b/assignments/week3/main.pdf index c7c94b59c01b030398e8f6e2036999c213552a69..ec7e2a09c2384ca0724d0e32ce72ecb78a0a05dd 100644 GIT binary patch literal 367776 zcma%?Q+F;*)MaBQC$^oO*tTukwr!o*wx8IxZQHilZ};f$s&9Ha2wl^__;p2mGc5yN_w1x55*v4LW!Wq5C)33MJR12Bkg0xj7t5DK$E73_DhU}#2 zYyc5(BEeWX}pKWK}qX8QClf+l(WKO?1KkxKLXJ?1E>&e&;KIN}5qyMci6-F%- zxjVV&P;62ZA{rENZnp8W+jV*U{Ktvb_Wg2@2DsgFnn+lPT03@*hEtr*rV@d=C-m_4`LGfTFxot%IFx5sub#1fmbnCD@$!&Wq;(BIkC`L&%#0+WW|j=@3jM=HvSAy5isP%hOBFA$WG zp1=-*GfQNTCajEMgQk0d;^d{K?bzub*j3wkA8#}UGD$!fplxch%3m?68{oc9*XkX}(tJnLJW5!p3 z>yiNFv$~gG<#g&dtpn!c%dIKXlOgNfg55r++BVAwYbf3mU4$APNtIFzcc8y;KMy|s z?Yj9m+(NxuIXH8w2n{6+YECi0zmkz?H%`HUKqgJ0^ALy8kkBJ+$%n; zk6`llT+Rb%I;)-gr}kU$TLs;Dk40-|w}1QX%eUQT_2ARs2x!esIELSaBbb(p_Ak>@ zR_Ak)|E@d1;BMbaT>^8@=M8!Es;>^y6&(t{z&9#6CpTE?n1lgj8wgY)_-?<67*H&x 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z$`V(j+vYK}F4-Z7N#Kwy%9U);*(@r&!i6XDAz4pKOz+Q~tIsFrtV>IdPzo2Fv%!1Y z06*IbQ~cvW@~(*eB#EFKuWeg=5h~j)XUg0!ua-EO(?7CB-_4i5AnAa!;Qj-7X#aaq;}` zLXt{Xh=Y$ED8b7k$sx|g!w2M+<`Nf|l#pTva`As*=LB*AMX3INm$y>@|3OG#+KDF; z1B84+0W(kqTXOnP`iUx1RBTG0&z1ZWx@>w4wfXT-P7RvRouX2?dRY;jr6a?t1SU%S z!#RN^8U_*H@dhR;lqzJbhUrzC1R9n{ w=H2>KY~@RA@%yg-#O8lOL;BzOz}dyf$;HD7YzE)}^6&$>0d#ay%F=-U0>#$gn*aa+ diff --git a/assignments/week3/main.tex b/assignments/week3/main.tex index fa368ff..f196ad8 100644 --- a/assignments/week3/main.tex +++ b/assignments/week3/main.tex @@ -1,8 +1,34 @@ \documentclass[../main_text.tex]{subfiles} \begin{document} +\setcounter{exercise}{0} -\section{Assignment 3} +\part{Assignment 3} \exercise* +\begin{tcolorbox} + Suppose $x,y \in \R$ and $x < y$. Prove that there exists $i \in \R\backslash\Q$ such that $x < i < y$. +\end{tcolorbox} + +If either $x$ or $y$ (or both) are not rational numbers, we can simply take the average like so: $\frac{x+y}{2}$, +in a similar way we did for the rationals. Since $x$ or $y$ isn't rational, the resulting fraction will also not be +a rational and this proves the statement. + +Now if $x,y \in \Q$, we cannot use this average trick, because the resulting fraction will be a rational itself and +so it doesn't satisfy the restriction that it must be in $\R\backslash\Q$. So we have to take a different approach. + +Let $x,y \in \Q$ with $x < y$ and $m := \frac{x+y}{2}$, so $x < m < y$. Then, let $X = \{a \in \R : x < a < m\}$ and +let $Y = \{b \in \R : m < b < y\}$. Since $x < m$ and $m < y$, these are nonempty and they are bounded, because +of the restrictions $x < a < m$ and $m < b < y$. So, there exists $k \in X$, that is not rational such that +$x < k < m$ and there exists $h \in Y$ that is not rational such that $m < h < y$. Pick either $k$ or $h$ as $i$, +since $x < k < m < h < y$. \qed + +\exercise* +\begin{tcolorbox} + Let $E \subset (0,1)$ be the set of all real numbers with decimal representation using only the digits 1 and 2: + \begin{equation} + E := \{x \in (0,1) : \forall j \in \N, \exists d_j \in \{1,2\} \text{ such that } x = 0.d_1d_2...\} + \end{equation} + Prove that $|E| = |\mathcal{P}(\N)|$. +\end{tcolorbox} \end{document} From d708244e55d13292c8230065f0c8bec9ab570cd8 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Tue, 30 May 2023 17:05:42 +0200 Subject: [PATCH 3/7] WIP on exercise 4, very close --- assignments/main_text.pdf | Bin 607015 -> 617785 bytes assignments/week3/main.pdf | Bin 367776 -> 472858 bytes assignments/week3/main.tex | 98 ++++++++++++++++++++++++++++++++++++- 3 files changed, 96 insertions(+), 2 deletions(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index ea9d43324fe2c890bbe3176554a58a0465f3d727..7011f7c05808f5eda7b5ce8709a90b5b3f50bcd0 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|\mathcal{P}(\N)|$. \end{tcolorbox} +As a hint to this exercise: \textit{Consider the function $f : E \rightarrow \mathcal{P}(\N)$ such that if $x \in E, +x = 0.d_1d_2...$, +\begin{equation*} + f(x) = \{j \in \N : d_j = 2\}. +\end{equation*}} + +In order to prove that 2 sets are of equal cardinality, we need to prove that there is a bijective function between +the 2 sets. In this case, the aforementioned hint function does the trick. Non-formally speaking, it is exactly what +we are looking for: it is a (weird) representation of the power set of natural numbers, in that for every decimal, +represented by a natural number, it is decided if that decimal is a 2 or a 1. This is similar to the actual power set +of the natural numbers, in which for every natural number it is decided whether the number is in a subset or not. + +Now for a formal proof. To show that $f$ is bijective, we need to show that it is surjective and injective. +\begin{description} + \item[Injectivity of $f$] In order to show that $f$ is injective, we have to show that for every $x \in E$, there + is a unique $y \in \mathcal{P}(\N)$ for the function, by showing that $f(a) = f(b) \implies a = b$. + + So, let's assume that for some $a,b \in E$, $f(a) = f(b)$. So, there two sets of natural numbers + $\{a_1,a_2,...,a_n\} = \{b_1,b_2,...,b_m\}$. Equality in sets means that every element that is present in the + one set, is present in the other, and vice versa. No element that is present in either set, is missing in the + other. So, in this case, both sets will represent the same sequence of digits that are 2. Because the only + other option for digits is 1, that means the complete digital representation of $a$ and $b$ are known, + unique and the same. This concludes the proof for injectivity. + \item[Surjectivity of $f$] To prove surjectivity, we need to prove that for any arbitrary $y \in \mathcal{P}(\N)$, + there exists a corresponding $x \in E$ such that $f(x) = y$. + + So, take an arbitrary $y = \{y_1, y_2, ... , y_n\}$, where each $y_i \in \N$ and thus $y \in \mathcal{P}(\N)$. + Then, corresponding $x \in E$ can be constructed easily as follows. Take a decimal number $0.d_1d_2...$ and + turn every decimal $d_i$ for which $i \in y$ into a 2, and every other decimal into a 1. Since every decimal + can only be a 1 or 2, this handles every decimal correctly. Also, $f(x)$ will be in $\mathcal{P}(\N)$. +\end{description} + +Since, $f$ is 1-to-1 and onto, $f$ is bijective. Then, because there exists a bijective function from +$E$ to $\mathcal{P}(\N)$, $|E| = |\mathcal{P}(\N)|$. \qed + +\exercise* +\begin{tcolorbox} + \begin{enumerate}[label=\emph{(\alph*)}] + \item Let $A$ and $B$ be two disjoint, countably infinite sets. Prove that $A \cup B$ is countably infinite. + \item Prove that the set of irrational numbers, $\R\backslash\Q$, is uncountable. You may use the facts + 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] + \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 + $f : A \rightarrow \N$ and let $g$ be the bijective function such that $g : B \rightarrow \N$. Then, we can + define a new function $h : A \cup B \rightarrow \Z$ as + \begin{align*} + h(x) &= f(x) \text{ if } x \in A \\ + &= -g(x) \text{ if } x\in B. + \end{align*} + Since $A \cap B = \emptyset$, this function is unambiguously defined. Since $\Z$ is countably infinite, + $A \cup B$ is countably infinite as well. \qed + + \item Because of part (a), we know that if we have two disjoint, countably infinite sets and join them, the result + is still countably infinite. The opposite must then also be true: if we have a countably infinite set and + we divide it into two disjoint subsets, both of which are infinite, then they still must be countable. + + So then, for $\R\backslash\Q$, we know that $\R$ is uncountably infinite. So when we split it into rational + and irrational subsets, from which we know that $\Q$ is countably infinite, $\R\backslash\Q$ must be at least + and at most uncountably infinite. \qed +\end{enumerate} + +\exercise* +\begin{tcolorbox} + Let $A$ be a subset of $\R$ which is bounded above, and let $a_0$ be an upper bound for $A$. Prove that $a_0 = + \sup A$ if and only if for every $\epsilon > 0$, there exists $a \in A$ such that $a_0 - \epsilon < a$. +\end{tcolorbox} + +Let $A \subset \R$, with $A$ bounded above by $a_0$. +So, we have to prove the implication both ways. First, let's prove that the implication to the right $(\rightarrow)$. + +Assume that $a_0 = \sup A$, so for all $a \in A$, $a \leq a_0$. Also, let $\epsilon > 0$. If $a_0 \in A$, +then we pick $a_0$ as $a$ and get $a_0 - \epsilon < a_0$, which holds $\forall \epsilon > 0$. If $a_0 \notin A$, +then we choose $a$ as the average of $a_0$ and $a_0 - \epsilon$, which is definitely smaller than $a_0$. We are +allowed to pick this as $a$, because we assume without loss of generality that $a \geq \inf A$. Then we get +\begin{align*} + a_0 - \epsilon &< \frac{a_0 + a_0 - \epsilon}{2} \\ + &< a_0 - \frac{\epsilon}{2} \implies \\ + -\epsilon &< -\frac{\epsilon}{2}. +\end{align*} +Since $\epsilon > 0$, this always holds. + +Now for the implication to the left $(\leftarrow)$. + +Assume now that the right side is true, i.e. let's assume that $\forall \epsilon > 0$, there exists $a \in A$ such that +$a_0 - \epsilon < a$. Again, let us first investigate the case where $a_0 \in A$. Well certainly still, if $a_0$ is +an upper bound for $A$ and it is also part of the set itself, it must be the supremum. Then, let's assume that +$a_0 \notin A$. Now, for all positive $\epsilon$, we know there exists an $a \in A$ such that $a \neq a_0$ and +$a_0 - \epsilon < a$. So, $a_0 < a + \epsilon$. $a_0$ is an upper bound and we can find $a$'s such that $a+\epsilon$ +is always bigger than $a_0$, so $a_0$ must be the supremum. If $a_0$ wasn't the supremum, then there must be some $b$ +such that $a < b < a_0 < a + \epsilon$. WIP \end{document} From 1639753ec0dc36fc919a7b23c681baa49a9c5d64 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Thu, 1 Jun 2023 23:51:07 +0200 Subject: [PATCH 4/7] Finished exercise 4 of week 3 and quite satisfied with it --- assignments/week3/main.pdf | Bin 472858 -> 479436 bytes assignments/week3/main.tex | 48 +++++++++++++++++++++++++++++++++---- 2 files changed, 43 insertions(+), 5 deletions(-) diff --git a/assignments/week3/main.pdf b/assignments/week3/main.pdf index 118e75df7ae4c739fddbab89b8327ef89d1b8d98..3e89b74c564f417b06c80932b9da6ddf5882d4a3 100644 GIT binary patch delta 310805 zcmV(#K;*xg?i|da9T6o^L`E$!E;R}z(A34)3NtdX5y}q%H?!OibPxeHkr5LEG%zxk zu?z<&f4y5-bK}Mle%G%+>q;t>%Enuj2d^Kv6eXcdky?RVdH;RB zIRFfxAwV7L2Ni(P96ddKeBC`H#dSf7AJ0g9{_XwQo3~aKR9B2jQM|t@G%K{!mF2Q{ ze_8xo{_xxS{@j$Sx>=tum?Wh*|KheRx+2U9-nVXCtF5@!I_WDU)-hDDeu1Q^rNPbEx5o80X=6uKw}dCU zebYsaF=1^o7`w-G#P$m$O({3~)`Xi~V^3APL`Qf>O52if zwQa&&?F?;G`y96);qr*PXlW&#N>GFcf2%a6fw0qfcH#W`_tkAUuaUa7^DK<^6D>5R 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zt5AW0iuY{SSW|s!+A6B^^=@ljeff&54Ryl-TP1ZPFXmTc4MQSoNd6j9K|>~M$bA~T zS3@CgC(hv-6>i`b?%*CC;8ES;o0+J7$Um#VBO{kNUI!fvH#j*kGYTaoMNdWwA)`#J diff --git a/assignments/week3/main.tex b/assignments/week3/main.tex index 0fcab65..ffa439b 100644 --- a/assignments/week3/main.tex +++ b/assignments/week3/main.tex @@ -121,7 +121,7 @@ Now for the implication to the left $(\leftarrow)$. Assume now that the right side is true, i.e. let's assume that $\forall \epsilon > 0$, there exists $a \in A$ such that $a_0 - \epsilon < a$. Again, let us first investigate the case where $a_0 \in A$. Well certainly still, if $a_0$ is an upper bound for $A$ and it is also part of the set itself, it must be the supremum\footnote{Proven in -earlier exercise}. +earlier exercise.}. Then, let's assume that $a_0 \notin A$. Now, for all positive $\epsilon$, we know there exists an $a \in A$ such that $a \neq a_0$ and $a_0 - \epsilon < a$. Let us assume then that this implies that $a_0 \neq \sup A$ and @@ -136,11 +136,11 @@ Since the implication holds both ways, the equivalence is proven. \qed \begin{tcolorbox} \begin{enumerate}[label=\emph{(\alph*)}] \item Let $a,b \in \R$ with $a < b$. Prove that the sets $(-\infty, a), (a,b)$ and $(b, \infty)$ are open. - \item Let $A$ be a set (not necessarily a subset of $\R$), and for each $\lambda \in A$, let $U_\lambda \subset - \R$. Prove that if $U_\lambda$ is open for all $\lambda \in A$ then the set + \item Let $\Lambda$ be a set (not necessarily a subset of $\R$), and for each $\lambda \in \Lambda$, + let $U_\lambda \subset \R$. Prove that if $U_\lambda$ is open for all $\lambda \in \Lambda$ then the set \begin{equation*} - \bigcup_{\lambda \in A} U_\lambda = - \{x \in \R : \exists \lambda \in A \text{ such that } x \in U_\lambda\} + \bigcup_{\lambda \in \Lambda} U_\lambda = + \{x \in \R : \exists \lambda \in \Lambda \text{ such that } x \in U_\lambda\} \end{equation*} is open. \item Let $n \in \N$, and let $U_1,...,U_n \subset \R$. Prove that if $U_1,...,U_n$ are open then the set @@ -157,9 +157,28 @@ Since the implication holds both ways, the equivalence is proven. \qed 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. + Let $a,b \in \R$ such that $a < b$. We want to show that for all $x \in (a,b)$ there exists $\epsilon > 0$ + such that $(x - \epsilon, x + \epsilon) \subset (a,b)$. Since for all $y \in (a,b)$ such that + $x - \epsilon < y < x + \epsilon$ this statement will recursively hold, we only need to prove that there + exists $\epsilon > 0$ such that $a < x - \epsilon$ and $x + \epsilon < b$. Then we can pick a fitting + $\epsilon$ in the following way, depending if $x$ is closer to $a$ or to $b$, formalized as follows. - \item - \item + If $x - a = b - x \implies 2x = b + a \implies x = \frac{b+a}{2}$, then $x$ is precisely between $a$ and $b$ + and we can pick $\epsilon$ to be $\frac{b - a}{4}$. Then $x + \epsilon < b$ and $x - \epsilon > a$. + + When $x - a < b - x$, then $x$ will be closer to $a$ then to be and $\epsilon$ is bounded more by $x$'s + proximity to $a$ than to $b$, i.e. $\epsilon < x - a$. So we can pick $\epsilon = \frac{x-a}{2} < x - a$. + Then $x - \epsilon = x - \frac{x - a}{2} = \frac{x + a}{2}$. Since $x > a$, $\frac{x + a}{2} > a$. For the + other side, $x + \epsilon = x + \frac{x - a}{2} < x + \frac{b - x}{2} = \frac{x + b}{2} < b$ since $x < b$. + So for both sides, we have shown that there exists an $\epsilon$ such that both + $x - \epsilon, x + \epsilon \in (a,b)$. All elements inbetween $x - \epsilon$ and $x + \epsilon$ will also + definitely be in $(a,b)$. + + The argument when $b - x < x - a$ is very similar and will be left to the reader. Then, for all $x \in (a,b)$, + the statement is proven. \qed + \item Non-formally speaking, in this exercise we want to prove that any union of open sets in $\R$ is open itself. + \item Similarly to the previous exercise, non-formally speaking we want to prove that any intersection of open sets + in $\R$ is open itself. \item \end{enumerate} From db44eae2be96350316caffc5ba89f2825e58f4e2 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Tue, 6 Jun 2023 20:49:45 +0200 Subject: [PATCH 6/7] Finished exercise 6 from week 3, just a few to go --- assignments/week3/main.pdf | Bin 482914 -> 487896 bytes assignments/week3/main.tex | 33 ++++++++++++++++++++++++++++++++- 2 files changed, 32 insertions(+), 1 deletion(-) diff --git a/assignments/week3/main.pdf b/assignments/week3/main.pdf index 5fdd3b841ad797fb06cbdf6c59dae95124debc66..adf7a1df6dc643c4a575e54505657b9a55722a50 100644 GIT binary patch delta 318176 zcmV)BK*PV{ydBu%9T6o^L`E$!E;R}z(A34)3Ntpb5y=k%H8Hc?4|5O#HZZeMAe;>s 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Then, for all $x \in (a,b)$, the statement is proven. \qed \item Non-formally speaking, in this exercise we want to prove that any union of open sets in $\R$ is open itself. + In order to make this formal, we will assume that $U_\lambda$ is open for all $\lambda \in \Lambda$ and + follow the definition as presented. + + So, let us assume that $U_\lambda$ is open for all $\lambda \in \Lambda$. This means that for every $x_\lambda + \in U_\lambda$, there exists an $\epsilon > 0$ such that $(x_\lambda - \epsilon, x_\lambda + \epsilon) \subset + U_\lambda$. To prove that $\bigcup_{\lambda \in \Lambda} U_\lambda$ is open, we need to show that the same + property holds for all $y$ in this set. But since the union between some sets is defined as the set that holds + all the elements that any of these sets hold, this is trivial: for any $y \in \bigcup_{\lambda \in \Lambda}$ + for which we want to know what $\epsilon$ we need to show that the union is open around that $y$, we just pick + the corresponding $\epsilon$ for the subset $U_\lambda$ which was open. Since all elements in that $U_\lambda$ + are also in the union, this must certainly be the case. \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. + 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. + + Formally, let $n \in \N$ and let $U_1,...,U_n \subset \R$. We assume that all $U_1,...,U_n$ are open. + \item \end{enumerate} +\exercise* +\begin{tcolorbox} + Prove that + \begin{equation*} + \lim_{n \rightarrow \infty} \frac{1}{20n^2 + 20n + 2020} = 0. + \end{equation*} +\end{tcolorbox} +In order to prove that this limit holds, we need to show that a function $\{x_n\}$ converges to $x$, +i.e. if for all $\epsilon > 0$, $\exists M \in \N$ such that $\forall n \geq M$ the following inequality holds: +$|x_n = x| < \epsilon$. + +Let $\epsilon > 0$. We choose $M \in \N$ such that $\frac{1}{M} < \epsilon$ (Archimedean Property). Then for all +$n \geq M$, $|\frac{1}{20n^2+20n+2020} - 0| = \frac{1}{20n^2 + 20n + 2020} \leq \frac{1}{n^2 + n} \leq +\frac{1}{n} \leq \frac{1}{M} < \epsilon$. \qed + \end{document} From 8d53e6e409fcdc17633b271c203a652c713847d4 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Fri, 9 Jun 2023 10:48:40 +0200 Subject: [PATCH 7/7] 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 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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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Again, let us first investigate the case where $a_0 \in A$. Well certainly still, if $a_0$ is -an upper bound for $A$ and it is also part of the set itself, it must be the supremum. Then, let's assume that -$a_0 \notin A$. Now, for all positive $\epsilon$, we know there exists an $a \in A$ such that $a \neq a_0$ and -$a_0 - \epsilon < a$. So, $a_0 < a + \epsilon$. $a_0$ is an upper bound and we can find $a$'s such that $a+\epsilon$ -is always bigger than $a_0$, so $a_0$ must be the supremum. If $a_0$ wasn't the supremum, then there must be some $b$ -such that $a < b < a_0 < a + \epsilon$. WIP +an upper bound for $A$ and it is also part of the set itself, it must be the supremum\footnote{Proven in +earlier exercise}. + +Then, let's assume that $a_0 \notin A$. Now, for all positive $\epsilon$, we know there exists an $a \in A$ +such that $a \neq a_0$ and $a_0 - \epsilon < a$. Let us assume then that this implies that $a_0 \neq \sup A$ and +try to come to a contradiction. So, then there must be some $b = \sup A$, which has as consequence that +$a < b < a_0$, since $b$ is still an uppoer bound of $A$ (and $a_0 \notin A$). Then, since $b > a$, we can pick +$a = b - \epsilon < b$. So, from our initial assumption we get $b - \epsilon < a_0 - \epsilon < b - \epsilon \implies +b < a_0 < b$, which is a false statement. So, $a_0 = \sup A$. + +Since the implication holds both ways, the equivalence is proven. \qed + +\exercise* +\begin{tcolorbox} + \begin{enumerate}[label=\emph{(\alph*)}] + \item Let $a,b \in \R$ with $a < b$. Prove that the sets $(-\infty, a), (a,b)$ and $(b, \infty)$ are open. + \item Let $A$ be a set (not necessarily a subset of $\R$), and for each $\lambda \in A$, let $U_\lambda \subset + \R$. Prove that if $U_\lambda$ is open for all $\lambda \in A$ then the set + \begin{equation*} + \bigcup_{\lambda \in A} U_\lambda = + \{x \in \R : \exists \lambda \in A \text{ such that } x \in U_\lambda\} + \end{equation*} + is open. + \item Let $n \in \N$, and let $U_1,...,U_n \subset \R$. Prove that if $U_1,...,U_n$ are open then the set + \begin{equation*} + \bigcap_{m=1}^n U_m = \{x \in \R : x \in U_m \text{ for all } m = 1,...,n\} + \end{equation*} + is open. + \item Is the set of rationals $\Q \subset \R$ open? Provide a proof to substantiate your claim. + \end{enumerate} +\end{tcolorbox} + +\begin{enumerate}[label=\emph{(\alph*)},wide] + \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. + + + \item + \item + \item +\end{enumerate} + \end{document} From 5930d51114ad0a38adfe405ba375b310ac608fac Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Fri, 2 Jun 2023 17:00:19 +0200 Subject: [PATCH 5/7] Finished part of exercise 5, getting more comfortable with these kinds of proofs now --- assignments/week3/main.pdf | Bin 479436 -> 482914 bytes assignments/week3/main.tex | 33 ++++++++++++++++++++++++++------- 2 files changed, 26 insertions(+), 7 deletions(-) diff --git a/assignments/week3/main.pdf b/assignments/week3/main.pdf index 3e89b74c564f417b06c80932b9da6ddf5882d4a3..5fdd3b841ad797fb06cbdf6c59dae95124debc66 100644 GIT binary patch delta 165277 zcmV()K;OU2p&jD99k6B~0X35WE-15)ATA9ZK7pURUH|w>zQ0!jWnoZi5%qu%#zIQ| z4|;pI29wt^e}cg662Nt?7OkOV1;pqp%wha^R1 zTV4+$OB;_iw)^!xhvaa`F=s~JbprHZSECUx=YHd%>Eg|TF1|XW@$+x5&p!Ur=>;={ zGbIY=`)-M~%*h2KjtVUooKs?~QZ`NP-&`!tq%?$bCq?#php!zG#+ZDxRE%)TB+quq z-$h7p(e$?4bCat?rq`aPmMABk%jW|LU`gt-UGjGs 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