From c9afa10ced55d31dd319da39622344c5f473cbac Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Sat, 13 May 2023 01:53:03 +0200 Subject: [PATCH] Finished 2 more exercises for week 1 --- assignments/main_text.pdf | Bin 528315 -> 542007 bytes assignments/main_text.tex | 1 + assignments/week1/1.pdf | Bin 440279 -> 453959 bytes assignments/week1/1.tex | 64 ++++++++++++++++++++++++++++++++------ 4 files changed, 56 insertions(+), 9 deletions(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index 48994540571a42ec57abb62b4f541910638540af..9ebb266c79ced20653da0aa867b8e8a64748d067 100644 GIT binary patch delta 291441 zcmZUaQ*fY7w6!O;ZQHhOO>Eod8(S0Gwmr$jwrwX9ClmkQsZ;0buYI$2_eI}zt$J#$ z?tJ3*Xc80(WpPOc7Df(Oin*oXHCR@*Unpo`+-zLwjfl|Hz_qTnBc2p$z-oQQD2Y+q zyC5iigGZuSvvk9+;;^8&2Vu1oc4nFhQjFp6fIh*KCHxd|svRSulJm;*y1J_W8Ha@{ z*5Y+fDz}5zx2vP$_SmRcwIUA9^>vAn9P-%CDdmXr>6w{k!+WB>{OPdqSqs-XtFQWX zy<@|^I3~bRBgWnCcNR;YA=65}#ne%^3D^76RX&P9vbglLfR8auF=e)#NQFDfyHazf 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b/assignments/week1/1.tex index 0b37136..0465667 100644 --- a/assignments/week1/1.tex +++ b/assignments/week1/1.tex @@ -4,11 +4,16 @@ \section*{Week 1} \exercise*[0.3.6] -\begin{enumerate}[label=\emph{\alph*)}] - \item Wanting to show: - $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ +\begin{tcolorbox} + Prove: + \begin{enumerate}[label=\emph{\alph*)}] + \item $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ + \item $A \cup (B \cap C) = (A \cup B) \cap (a \cup C)$ + \end{enumerate} +\end{tcolorbox} - In order to prove this equivalence, we have to prove the implication both ways. We use two lemmas for this. +\begin{enumerate}[label=\emph{\alph*)}, wide] + \item In order to prove this equivalence, we have to prove the implication both ways. We use two lemmas for this. \begin{lemma} \label{lem:set1} @@ -28,6 +33,7 @@ \label{lem:set2} $(A \cap B) \cup (A \cap C) \implies A \cap (B \cup C)$ \end{lemma} + Let $x \in (A \cap B) \cup (A \cap C)$. By the definition of set union, $x \in (A \cap B)$ or $x \in (A \cap C)$. By the definition of set intersection, $(x \in A$ or $x \in B)$ and $(x \in A$ or $x \in B)$. @@ -38,12 +44,52 @@ Using Lemma \ref{lem:set1} and \ref{lem:set2}, we get the desired equivalence of $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$. \qed - \item Wanting to show: - $A \cup (B \cap C) = (A \cup B) \cap (a \cup C)$ - - This proof is so similar to a) that it feels like a waste of time and will therefore be left to the reader. + \item This proof is so similar to a) that it feels like a waste of time and will therefore be left to the reader. \end{enumerate} -\hrule + +\exercise[0.3.11] +\begin{tcolorbox} + Prove by induction that $n < 2^n$ for all $n \in \N$. +\end{tcolorbox} + +For this proof we will use induction. For this, we have to prove the base case, i.e. $n = 1$, +and the inductive step, $n < 2^n \implies n + 1 < 2^{n+1}$. + +First, let's prove the base case. When $n = 1$, we get $1 < 2^1$, which is certainly true. + +Then, for the inductive step. We assume that the proposition holds for any $m \in \N$. +So, $m < 2^m$. Multiplying both sides with 2 gives $2m < 2^{m+1}$. +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} + +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. +It is built up by all the possible combinations of the different elements being either inside a certain subset or not. +For all possible subsets of $A$, we have that for every element $x \in A$ there are 2 possibilities, +either $x$ is in the subset or it isn't. +This means that for every additional element, the number of subsets increases by a factor of 2, with a minimum of 1, +in case of $A = \emptyset$. We will prove this formally now, using induction. + +For this, the base case is a set of 1 element (but the theorem also holds for the empty set, where $n=0$). +Let us assume that $A := \{ \pi \}$. +Then the cardinality of $\mathcal{P}(A)$ is $2^1$, with $\mathcal{P}(A)= \{ \emptyset, \{ \pi \}\}$. + +For the inductive step, we assume that for any set $B$ of cardinality $m$, the cardinality of the power set of $B$ +is $2^m$. Then, we will add an element $x \notin B$ to $B$ to increase its cardinality by 1, to $m + 1$, +creating a new set $C$. +Note that all the possible subsets of $B$ are still viable subsets of $C$, since $B \subset C$. +In order to create the new subsets, we can simply keep all the subsets of $B$, duplicate them and take the union with +the new element $x$, so now we also have all combinations of the old sets with possibly $x$ being in them. +Since this doubles the number of subsets, the cardinality of $\mathcal{P}(C)$ is $2^{m+1}$. + +Both the base case and the inductive step hold, which closes the induction and proves the proposition. \qed \end{document}