From d708244e55d13292c8230065f0c8bec9ab570cd8 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Tue, 30 May 2023 17:05:42 +0200 Subject: [PATCH] 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 100644 GIT binary patch delta 249463 zcmV)7K*ztQh9$Y$C9w1X0XUao5D6)Nb&a780x=K;eZOL5E85$>cH6Bbxg$Z>5fmH* zdl)_+2;>3+^~}5(rVBA`4S#mi%`j=97(){Q^)ppal^R+S>L=RU@jfq3?eTh>orkbZ z&L>2LjmaJQ?&f886~ZW}lD-Cv=%9b+SAcqiQ!*bA~nibauZ{LrFlv13b$r?Di*BggUI z?>W2;Ihv7_Hd|o#!I&9}JUl%23ti9N&Fa~=XLWe~?egsTKa84DO&FD8cDb5qHq%lQ z%jN8HG5e`{)!gsS=S(apW>`Vw)&8*=4=q-){5Uq7KS4e!Lg7q6@LjeT@m7%^=Y7>Wg2y~G(=iqZMGN=e-Gel+i#E!WwcC#IW7B{0jYk6bh?B| z9LZ=JhX7WD5Qoya2jdeo9?%v>qLfn!7{P-P4PZdfY2cjaUjKf*_S_oic7!uG+5ws? z&@!zMl2d!&Ed?QvlA$7oNnwst#5WIjcjt3mZ9x&8>e5>0O9UrM!4_`vr}Me8)mP_p zf2wr#6z7Y4J_C8uA})B#X|M>Cs(hLy4A>B6{mV}&aiujeoW?25W-*E$Zf9tQ9~Zj% z8cd2d5y28HNDXLLm``A$Wz{P*McZ>-eS~MQN4uM57vG$z3gb(d_Nnc5-6%Sg8bZM| z1^DeKi4>7FJ&8hQK%oI3O$=Is0Eluie+ozurV|HWkCK|^7Kk z@o|_3Abw~m6BE$#4+T5(1f|r1WWVn9XC4%2-fPfnr*)fIF%7)QJaZ=_x)B8_`W`Ak zx8avHp!jly1d-LUb;2;p`KZv86AFkF>pzWAA>WWPuq51+tF}jW;u5WUx#sEUe@;Sz z0u0XgwwCod)wrWO=M`vpU2S(a@B{TjP0|5bZMHlAa^LLQ?WVa6!w+k}aCzT7^FxpQ zqU&bkU%GaEcl+eucg_3oGY~dx4$Ui1V33Dh;s=^<-HtO9y~O#qKs zP#zVhUm*zRNhAI|22&EMk`a4~f1$~Tv%S3_%0U;jKKtqCdbWVyz5{9t@YauR<{G#} zj6{QTJ9~5X+6!KZjResJCk?P_B4vn=*fBmzhQdrsK0d=$Zko*`xrPu4;JX>EV*;-l z{{UXJw~gbvs2-aqKbpA4t3|u;FDow_JTgio=6BADs_IR>ryY5XHsw%R$wV5Hm5zugsXtV`a@zjV81 zx5Pl5cw8Xv$l~7bQlKj@K?;8Kf8UEOBDsvh3ljiF zw|erkf>7vyF&;6Bt4@(m4{ZI=g^^4yHQJI3In zZb2&Q0seEjJE&xqe-bJOYfjNIO#aL%I>tCVS;vqNl$XdnLRfUzOejc@)2cioG zd8&{G@K%s~{m$?LJP?r_Ll!`v9||urax(>s>H)dY?-_%pf94?3jex*}A7b{}KTc#l z=Z3(%5+G`_vDM|xaud+HJ)%Y20;%W#%nwt-AVm^}Z##W+S7MT#sp%Lv~Tgdf)g5Eqeh!G9m&4VW@vEumEQ8aZc*9_0oG4h;4w zR{uDDXb1wBf6Tc-#WXj#NoT*B;0EQVWZ#xo#}z(g{2I66 zSPrb!soY?^22s-`QBhO4I3YS`3Opvnq{V8??G2Aeu8*l|4IfnO%f#VW9l~xg*taCr zQ4;||OF>2o5NH>aODtWW211!~Sa{!TH_P?i?kW5df8`b#nJ!yc>x6=-PXGVTH*@rE6uHK60OM7GF8_&QRIi=m(dsZfJ6 zzMnDXwoq%0xcOmK@E zb`rOeDgW`vc{s$kRP%E8z$G|>OA_H3cwq`&f9O~`W%@WHgl2?mQEpV~(9Q8mYKA!$ zge12mKr&2Zgp>rEN$%HIklBW+_Hn2}$=hJ}YX4k|@EQEDk=qNyTvGFZ4k4x^}Q+;(AMLPc%u6VcrjLM1mg$xVor$$ z+z%WM1po&6X+1A~q2s48uH%0CDcE@Zf)T;_@3?$$RH}lEPDm7_vMg0tE*4`IUF}R+ zPD47u4OSO49EB!bH8Y3CPJ?!FaWU0Bu?I^;5Y<9#4KY%o-;=jqf>iTCe^roYy#zH* zOk`z3<;5rqt;P`IVu(Oq*kr_$)G)t*X=X_bCH zigN*`M|p5RDJS8nS7S0?K5;iD1PzmSWBMR^BpApGzSOnI8EhEqkKQnTZY2ai|P@Vee_2UBOGqe;kyCiasXE|Bd z0{cU@4yD1DF3Dh6f3Gu@Go|>o+xk08;VzNsg9Dx}hAXxC4!D-#EZB~x_?p_vbnNALiy$i)jB|`CL&j1T|O=+*(%WAwOnC^9L=CB_mo-UUZ zw(8u~?`88I^2}#`$ldwDXXl@;Xy07#;*!VEv9O76@S4~U@3#)W?R%fP!T@ZE*_Wkz zu#rK+q}!(se_-pRB-}fNITP72{IXrej)~1qOfe)EB}mLC^INdWa|T~n18raf5%ZF-zu~l+`1cOlmj>EUCfl>sq4SFsv_4B6x(OQGM1dd8{)M9rRlB%N_i#Vh#|?+R zH@IpGVTmzi6x43^aVrxv-_FR~Z^?s~8GzQ^e>}j#-gwXqy3D06D41ym>T$KOUFnU{U?0{TM2G4(ZvRaZ7L{jv|CK|VN@jZr5p&5J? z6fX!+s<2PXz7j-nF|cMM$&mVtWT=^_dr2$zlA(#>s}f%rr>hB^@SiM)g{^w{uX;9< z89tt-GM?Y&ZzfVVOZQ5DdwG9W`nwvFHHZS|(nfu2e=cd?h`rmbb_{fU7>yEn(KGUMVuX{IoyFa+0ZVx~H>-w+nzG2j)`!T;6mx35= z^3Cc$TrQb5#Z6c$v!x}eSOu%E*D&`-!NlYa72%d7JL_vVWtL`cN>WxbN8{===A|S| z=zPiTB*um?t^+D$5~?Iz#VPe96h#!^@29*2{+hD*<1~tYCDBrbsc(bR%?wVW98fyV zV<9OqW+Ybx=qPnAUj6;%a9I``CrkY!Jny|YZF(vCDNU%VP_F{-ZXZ{IYOBPwKk5>K#q<;cTLwd?HzuRCup&l%EY=MM_RY4 zzYoBFR&}?D6HxTfgaM4m%g(KMY_?lBW|^6@G;S}8uHMCSxLlgzm*b{g`QP@|8=&Y> zIVPD>!W45PubaE3A4yeJ)WI&pD7VDX;)C5|RUf;Jo8<8Tqup-3+&PBWU-u7vvv1w7 z3lrdB*sY4f$gTpoAf}n1`^{GS-2;Rpn?MzR9}%TXMGRF5QPx=in@2pX16tbiXx{O2 zVa-eix<0GmoG2qki$KOt?xbKRJ|g1g+)f&dFC1>ZQ#?3b{M_!rFg&(i+>UJn0^**% z(&@lo?PRHPZn`Y4z07Lhd%M~m*PCvbZ%<&%euJM52g~a{3X65#iu&`1ho=3>Nm1W_ zd_qg7Di@N(i1pn!rB3Mu$+Pg6oDr=nXJo%W9&#`i_RMLT3IaZz#Yo9+v)pAG>;lsZ zEi14OURka$HMb#(t^$?DSt~czImGlppRvkrK&OXUt({^CSi6zl!Yosm#?6jO39}p* zVW@7O+(>=6Y5GGQO#dknlSU9-RXII>Pt(B@1=qwT;%*`HxVsWO%=&H;68MlIItZ~R z)3dSXw{!S#b29i$&P{=!RJ6?LyqcC-@Wo7e#YtjOkR|l0pPSB_D)?HvkJbL*%*cZk z?skoyVUp<(0c#e?`aHv&lB+k!Zcx5smed_r_YPSrH&+9_O+pBpdb5c(?|%ruCmM*B(v{$RN+b&uWi}e1YIu zsJY;oDN^b&(c|FRl)ol;ews-+Bz>Fc`DrG4t~md)l;e!cD=WxmMyKr0_i<2QINMDP zdT{DQAt@nB%$=w?c?Qjlw3iTnR3+0Aoz86@8Z=ssjNUBKq7o2_n;cW;zzs%C$L_S@ z7hoXd>qU3g?&sM5-xz*bf(@Y~HbFQeKsK)LH|^D22B0dy)@M2|x9|aof&9A6VEne$ zKCNXT0U&Z20oJ(D(kB-i_XUEuu0I@C5I!$U8+5?nrtTUWx5IPYxzF2w{dQ}E#?A(U z93mcce?NOTwfk7R;H%l$5E}-Pdd$gk44x5%vq>?glZ;SAsVHOb*HdN;-Pjk|#`Vj} z6fd-iU<}WkQxvokLdsI53XDat;{lJ1qvWjwJrtJ zWoFRkC5=m>QPNV_zV?fMmbW=$Y&isTB1gD4tqr-ku_6t; 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|\mathcal{P}(\N)|$. \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}