From c05675e16d222fe62da5205683b35552aea92042 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Sun, 14 May 2023 01:56:35 +0200 Subject: [PATCH] Finished all the exercises, except for the last one of week 1 --- assignments/main_text.pdf | Bin 542007 -> 551521 bytes assignments/week1/1.pdf | Bin 453959 -> 463392 bytes assignments/week1/1.tex | 58 ++++++++++++++++++++++++++++++++------ 3 files changed, 50 insertions(+), 8 deletions(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index 9ebb266c79ced20653da0aa867b8e8a64748d067..02d2c88aaeb7d07278661dcadbe6504bf51ff972 100644 GIT binary patch delta 390745 zcmZtNV~;LetT5oQZQFZn+qP}ny2rL{+r}Q-wr$Vs(RohJ$M;RzmHvPx?Uk&yM-d+^ z6H|~Yii*=Q(X+vj&o7Ux!?3UsF%mf#Tf^}3!Z661*;}|;60xwbq|d=XQ3GypZCs8< zZ@K~C0?NrtyM2V2u4SFFNoC1x4`wLtt7UV=E`U`NJ(1?#Iu|!|M}{n28{jA~ubY5w(FFvY5_6MacfPjt*LXn%{>V zQyN(>rc7tCDS%vA)z_t52f(X0qjqp59d&eSYvb}eB-6QjTi(C#-A8ba3|ZMll!4Z0 z8BK*6s+HdG*N8c_u1Bjb%>z|!m-tmo@Ph-z|M#gw_Zs8=-rvh7_x>^ndoVL8!ef$^ 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z)IHo;t*UF6{TDy|dY&IWFD=j8z308+`N()aZ|(R5&fpv_;1aIjTHRN{A9ot^4=!IJ l#tLO_WOH<KWnpa!Wo~3|mj)6C4h%IjG&VO1B_%~qMhaz=3PS(@ diff --git a/assignments/week1/1.tex b/assignments/week1/1.tex index 0465667..715613b 100644 --- a/assignments/week1/1.tex +++ b/assignments/week1/1.tex @@ -4,6 +4,7 @@ \section*{Week 1} \exercise*[0.3.6] +% For some reason I can't put a fitted tcbox here and I really don't like it \begin{tcolorbox} Prove: \begin{enumerate}[label=\emph{\alph*)}] @@ -48,10 +49,8 @@ \end{enumerate} -\exercise[0.3.11] -\begin{tcolorbox} - Prove by induction that $n < 2^n$ for all $n \in \N$. -\end{tcolorbox} +\exercise*[0.3.11] +\tcbox{Prove by induction that $n < 2^n$ for all $n \in \N$.} For this proof we will use induction. For this, we have to prove the base case, i.e. $n = 1$, and the inductive step, $n < 2^n \implies n + 1 < 2^{n+1}$. @@ -65,10 +64,8 @@ Since $m \geq 1$, $m + 1 < 2m$, and thus $m + 1 < 2m < 2^{m+1}$. Since both the base case and inductive step hold, we can close the induction, proving the proposition. \qed -\exercise[0.3.12] -\begin{tcolorbox} - Show that for a finite set $A$ of cardinality $n$, the cardinality of $\mathcal{P}(A)$ is $2^n$. -\end{tcolorbox} +\exercise*[0.3.12] +\tcbox{Show that for a finite set $A$ of cardinality $n$, the cardinality of $\mathcal{P}(A)$ is $2^n$.} The power set of a set $A$, $\mathcal{P}(A)$, is defined as the set of all possible subsets of $A$. This is very similar to an inclusion/exclusion problem. @@ -92,4 +89,49 @@ Since this doubles the number of subsets, the cardinality of $\mathcal{P}(C)$ is Both the base case and the inductive step hold, which closes the induction and proves the proposition. \qed +\exercise*[0.3.15] +\tcbox{Prove that $n^3 + 5n$ is divisible by 6 for all $n \in \N$.} + +In order to prove this proposition, we will use induction. To do this, we need to prove the following lemma, +of which we will see the usefulness later: + +\begin{lemma} + \label{lem:div6} + $3n^2 + 3n + 6$ is divisible by 6 for all $n \in \N$. +\end{lemma} + +This lemma we will also prove by induction. For this, we prove the base case and the inductive step. +First, for the base case we have $n = 1$, yielding $3 * 1^2 + 3 * 1 + 6 = 12$, which is divisibly by 6. + +Then, for the inductive step we assume that the lemma holds for a certain $m \in \N$. +So, $3m^2 + 3m + 6$ is divisible by 6. +Substituting $m$ with $m+1$ gives $3(m+1)^2 + 3(m+1) + 6$, which can be expanded to +$3m^2 + 9m + 12$. Rewriting this with our assumption in mind gives the following: +$(3m^2 + 3m + 6) + (6m + 6)$. We know from our assumption that the first part is divisible by 6, and since $m \in \N$, +$6m + 6$ is also divisible by 6, and so the whole expression is as well. \qed + +Now for the original proposition. We will prove this by induction. First we prove the base case, where $n=1$. +Then, $1^3 + 5*1 = 6$, which is definitely divisible by 6. + +For the inductive step, we assume that the proposition holds for a certain $m \in \N$. So, $m^3 + 5m$ is divisible by 6. +When we increase $m$ by 1, we get: $(m+1)^3 + 5(m+1)$. +Expanded, this is the same as $m^3 + 3m^2 + 8m + 6$. When we rearrange the terms we can get the following expression: +$(m^3 + 5m) + (3m^2 + 3m + 6)$. From Lemma \ref{lem:div6}, we know that the latter part is divisible by 6. +The prior part is divisible by 6 because of the assumption of the inductive step. So together, this expression +is also divisible by 6. \qed + + +\exercise*[0.3.19] +\begin{tcolorbox} + Give an example of a countably infinite collection of finite sets $A_1, A_2,...$, + whose union is not a finite set. +\end{tcolorbox} + +The easiest example is simply the collection of singleton sets containing a natural number. +So each set $A_i := \{i\} \forall i \in \N$. Since $\N$ is countably infinite, so the collection of sets. +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* + \end{document}