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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} -- 2.34.1 From c05675e16d222fe62da5205683b35552aea92042 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Sun, 14 May 2023 01:56:35 +0200 Subject: [PATCH 3/8] 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 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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} -- 2.34.1 From 5cb78813a529a2c6c516dad634fe43c8edba5acb Mon Sep 17 00:00:00 2001 From: Philippe Zwietering <philippe.zwietering@gmail.com> Date: Mon, 15 May 2023 00:02:46 +0200 Subject: [PATCH 4/8] Working on last exercise of week 1, almost done --- assignments/week1/1.pdf | Bin 463392 -> 470305 bytes assignments/week1/1.tex | 27 +++++++++++++++++++++++++++ 2 files changed, 27 insertions(+) diff --git a/assignments/week1/1.pdf b/assignments/week1/1.pdf index a65baf2c08eb1f29878471ab39a696e0309e5fc3..b5e39f4e9a5333d40a1fbe0a2ca3e15f362d8df4 100644 GIT binary patch delta 376114 zcmV(|K+(UTq#U8z9I!SI12H)=mk;9xD1Y4>Yj4~(^1FY9zMYKPjQA#dC~%jeDOwc4 z<zk>8`asv~bs}^hu6H+%|NYHyNJ`XdS9Tg8z<n?z&6306yk|%^)w`;xzCUZ?^X1jq 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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* +\begin{tcolorbox} + \begin{enumerate}[label=\emph{\alph*)}, wide] + \item Compute $f(4/15)$. Find $q$ such that $f(q) = 108$. + \item Use the \textbf{Theorem} to prove that $f$ is a bijection. + \end{enumerate} +\end{tcolorbox} + +See the assignment PDF for the full assignment specification and theorem. + +\begin{enumerate}[label=\emph{\alph*)}, wide] + \item $\frac{4}{15}$, if written as a product of prime factors, is equal to $\frac{2^2}{3^1*5^1}$. + Since this fraction is not a natural number, we have to use the second part of the definition of $f$. + So, $f(q) = 2^{2*2} * 3^{2 * 1 - 1} * 5^{2 * 1 - 1} = 240$. + + For the inverse of $f$, it is still necessary to compute the factorization in prime numbers. + Using the powers of the primes we can deduce whether the prime present is, if applicable, + part of either the numerator or the denominator. + $180 = 2^2 * 3^2 * 5^1$. Because of the way $f$ is defined, we know that all the prime factors with an even + power are part of the numerator and all prime factors with an odd power are part of the denominator (except 1, + which just maps to itself). When we backtrack using this information, we then get the following fraction: + $\frac{2^1 * 3^1}{5^1} = \frac{6}{5}$. + \item In order to prove that $f$ is a bijection, we have to prove that $f$ is injective and surjective. + \begin{description} + \item[Surjectivity:] + \item[Injectivity:] + \end{description} +\end{enumerate} \end{document} -- 2.34.1 From 04dbe12f83dd517d14504fe87c0284eae9bb1601 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering <philippe.zwietering@gmail.com> Date: Mon, 15 May 2023 17:08:39 +0200 Subject: [PATCH 5/8] Working on the injectivity of the case where x is a fraction of the last exercise --- assignments/week1/1.pdf | Bin 470305 -> 478953 bytes assignments/week1/1.tex | 52 ++++++++++++++++++++++++++++++++++++++-- 2 files changed, 50 insertions(+), 2 deletions(-) diff --git a/assignments/week1/1.pdf b/assignments/week1/1.pdf index b5e39f4e9a5333d40a1fbe0a2ca3e15f362d8df4..fc043cbfd8afcca5e8298b08001fb1170577e585 100644 GIT binary patch delta 314241 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injective and surjective. \begin{description} - \item[Surjectivity:] - \item[Injectivity:] + \item[Surjectivity:] We want to show that $f$ is 1-1, + i.e. $f(x_1)=f(x_2) \implies x_1 = x_2$. + + So, let's assume that for any $x_1, x_2 \in \{ q > 0 : q \in \Q \}$, $f(x_1) = f(x_2)$. + Since the function $f$ has 3 parts, based on the input, we have to prove this statement for those 3 + parts separately as well. First, the easiest case, where the input set is $\{1\}$. + Then, $f(x) = 1 \forall x$, so $f$ is injective. + + For the case where $x \in \N\backslash\{1\}$, $f(x) := p_1^{2r_1}***p_N^{2r_N}$. + We know from the \textbf{Theorem} that any fraction can be uniquely written as a product of prime + factors with exponents, so when we assume $f(x_1) = f(x_2)$, we can also assume that $x_1$ and $x_2$ + have a unique prime factorization associated with them. So let's assume that $f(x_1) = f(x_2)$. + This means that $p_1^{2r_1}***p_N^{2r_N} = q_1^{2s_1}***q_M^{2s_M}$, where $p_i^{r_i}$ and $q_j^{s_j}$ + denote the prime factors for both sides. We can further expand this expression into: + + \begin{align} + p_1^{r_1} * p_1^{r_1} *** p_N^{r_N} * p_N^{r_N} &= + q_1^{s_1} * q_1^{s_1} *** q_M^{s_M} * q_M^{s_M} \implies \\ + p_1^{r_1}***p_N^{r_N} * p_1^{r_1}***p_N^{r_N} &= + q_1^{s_1}***q_M^{s_1} * q_1^{s_1}***q_N^{s_M} \implies \\ + x_1 * x_1 &= x_2 * x_2 + \end{align} + + Because we know that each fraction constitutes a unique prime factorization, we also know that $x_1$ and + $x_2$ are uniquely derived. This is why the implications in the equation above holds. + Because both $x_1$ and $x_2 > 0$, $x_1 = x_2$. + + Now for the case where $x \in \Q\backslash\N$. + Then $f(x) := p_1^{2r_1}***p_N^{2r_N}q_1^{2s_1-1}***q_M^{2s_M-1}$, using the unique factorization + derived from the \textbf{Theorem}. So again, we assume that for any $x_1, x_2 \in \Q\backslash\N$, + $f(x_1) = f(x_2)$. + Using the definition of $f$, we get: + $p_1^{2r_1}***p_N^{2r_N}q_1^{2s_1-1}***q_M^{2s_M-1} = + v_1^{2t_1}***v_n^{2t_n}w_1^{2u_1-1}***w_m^{2u_m-1}$.\footnote{The super- and subscripts become a bit + abracadabra, but I think everything is unique and readable this way.} + Expanding this expression further, we get: + + \begin{align} + \frac{p_1^{2r_1}}{p_1}***\frac{p_N^{2r_N}}{p_N}\frac{q_1^{2s_1}}{q_1}***\frac{q_M^{2s_M}}{q_M} &= + \frac{v_1^{2t_1}}{v_1}***\frac{v_n^{2t_n}}{v_n}\frac{w_1^{2u_1}}{w_1}***\frac{w_m^{2u_m}}{w_m} + \implies \\ + \frac{p_1^{r_1}*p_1^{r_1}}{p_1}***\frac{p_N^{r_N}*p_N^{r_N}}{p_N} + \frac{q_1^{s_1}*q_1^{s_1}}{q_1}***\frac{q_M^{s_M}*q_M^{s_M}}{q_M} &= + \frac{v_1^{t_1}*v_1^{t_1}}{v_1}***\frac{v_n^{t_n}*v_n^{t_n}}{v_n} + \frac{w_1^{u_1}*w_1^{u_1}}{w_1}***\frac{w_m^{u_m}*w_m^{u_m}}{w_m} \implies \\ + \frac{x_1*x_1}{p_1***p_N*q_1***q_M} &= \frac{x_2*x_2}{v_1***v_n*w_1***w_m} + \end{align} + + \item[Injectivity:] We want to show that $f$ is onto, + i.e. $f(\{ q > 0 : q \in \Q \}) = \N$. \end{description} \end{enumerate} -- 2.34.1 From ad4aaab3ebabd5d8f5680a8b59065920a3e35f38 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering <philippe.zwietering@gmail.com> Date: Tue, 16 May 2023 15:35:53 +0200 Subject: [PATCH 6/8] Got stuck on final part of injectivity proof, finished all the rest --- assignments/main_text.pdf | Bin 551521 -> 570002 bytes assignments/week1/1.pdf | Bin 478953 -> 481922 bytes assignments/week1/1.tex | 34 ++++++++++++++++++++++++++++++---- 3 files changed, 30 insertions(+), 4 deletions(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index 02d2c88aaeb7d07278661dcadbe6504bf51ff972..84f1e19e6cd4f93a38d416cd81a63857aa81fc52 100644 GIT binary patch delta 384993 zcmZUZQ*bT{u%%<$wr$(CZQK51+s=;d?AYG1ZQJIYTQl=?tGc?^OTVtJT3_!k5hov$ z{J{rf<7DE_0Jesr0xaukyOMP{2b|WQabntYK)ec%@X)RE;#Mo!S+NzbLB`oFv*nms 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zT)F*Q{CY4yk)5Tm6chVLkLn0jW(*~^jw>k@&;bRB<u-t+76>ik1Zz+c3eXEgR0n{Z zDj~81U@Vp3-Ib6L_d+DjG{^)VtEjBPvDW@;>}VCH&lM<Y72<IgNDd=JeKiEjH4iUL z*%yhs1U`lmWPGNI_QBEqjS@y0DNKL{fFzemHZ0?VGX)??OASDj7^7Jfg19yyJ~lwY zD}~bA^^+YL!*UpFOShSw)ctyXGX7iDBtR^$qxyR*M4H?D(@<VV(*sb4P5f_x%Uqt} zNgLgFl|m)kOeiIQ`B?SW0OHml;g+r5HeJtx>pNRr>D5=g0A2bqv3Kt;%?xcFXhH-| QE>><1I7&)!MG3h71IJ&j;Q#;t diff --git a/assignments/week1/1.tex b/assignments/week1/1.tex index 7811909..05de978 100644 --- a/assignments/week1/1.tex +++ b/assignments/week1/1.tex @@ -156,7 +156,7 @@ See the assignment PDF for the full assignment specification and theorem. $\frac{2^1 * 3^1}{5^1} = \frac{6}{5}$. \item In order to prove that $f$ is a bijection, we have to prove that $f$ is injective and surjective. \begin{description} - \item[Surjectivity:] We want to show that $f$ is 1-1, + \item[Injectivity:] We want to show that $f$ is 1-1, i.e. $f(x_1)=f(x_2) \implies x_1 = x_2$. So, let's assume that for any $x_1, x_2 \in \{ q > 0 : q \in \Q \}$, $f(x_1) = f(x_2)$. @@ -180,7 +180,7 @@ See the assignment PDF for the full assignment specification and theorem. \end{align} Because we know that each fraction constitutes a unique prime factorization, we also know that $x_1$ and - $x_2$ are uniquely derived. This is why the implications in the equation above holds. + $x_2$ are uniquely derived. This is why the implications in the equation above hold. Because both $x_1$ and $x_2 > 0$, $x_1 = x_2$. Now for the case where $x \in \Q\backslash\N$. @@ -204,8 +204,34 @@ See the assignment PDF for the full assignment specification and theorem. \frac{x_1*x_1}{p_1***p_N*q_1***q_M} &= \frac{x_2*x_2}{v_1***v_n*w_1***w_m} \end{align} - \item[Injectivity:] We want to show that $f$ is onto, - i.e. $f(\{ q > 0 : q \in \Q \}) = \N$. + I'm kinda stuck at this point. I see that this is definitely injective, since the way the exponents + are defined, you will always know which prime factors belong to the numerator or to the denominator. + But I fail to prove this using the direct definition of $f$ like we could do for the natural numbers. + This is because the products of the denominators in the last equation are not unique. + So maybe I simplified them too much and shouldn't try and write them in terms of $x_1$ and $x_2$ like + we did earlier, and try and focus more on just the exponents, but I feel it becomes really hard + to show that $x_1 = x_2$ that way. + + \item[Surjectivity:] We want to show that $f$ is onto, i.e. $f(\{ q > 0 : q \in \Q \}) = \N$. + + In order to prove this, we will take an arbitrary $y \in \N$, and show that $\exists x : f(x)=y$. + We know from the \textbf{Theorem} that $y$ can be written as a product of unique prime factors, + $p_1^{r_1}***p_N^{r_N}$. From the definition of $f$ we know that if the exponents of the prime factors + $r$ are even, they belong to the numerator of $x$ and if the exponents are + odd, they belong to the denominator of $x$. If there are no prime factors with odd exponents, + $x$ will be a natural number. If $y=1$, $x=1$. + + We will now only consider the case that $y$ is a prime factorization with factors with odd exponents + \footnote{The case for a prime factorization with solely even exponents can be backtracked + in a similar fashion, just without the case for odd exponents and making $x$ a fraction.}. + Then, we can find $x$ in the following way: we multiply each prime factor $p_i^{2r_i-1}$ with $p_i$ + and take the square root. We know that the square root of $p_i^{2r_i}$ is defined, since the exponent + is multiplied by a factor 2, which the root negates. This will yield a prime factorization that we will + put in the denominator of a fraction. We do the same for the prime factors with even exponent, but + without multiplying with $p_i$. The prime factors we gain like that we put as a product in the + numerator of the fraction. So, we gain a fraction with both the numerator and the denominator + consisting of products of prime numbers, which are natural numbers, and so the fraction is positive + and in fact a fraction. \qed \end{description} \end{enumerate} -- 2.34.1 From 5b11d9202a0c3cc7609feff20113477c258d6557 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering <philippe.zwietering@gmail.com> Date: Tue, 16 May 2023 15:48:46 +0200 Subject: [PATCH 7/8] Changed my commentary on failure to be cursive --- assignments/week1/1.pdf | Bin 481922 -> 487434 bytes assignments/week1/1.tex | 5 +++-- 2 files changed, 3 insertions(+), 2 deletions(-) diff --git a/assignments/week1/1.pdf b/assignments/week1/1.pdf index 33735dc99ee2082e876f23738c9d71f3adcf5d29..257c4416d610766b77d92df35e90b6dfdb6bc826 100644 GIT binary patch delta 98669 zcmV(zK<2-KwH=D!9kAyz0y8<65FiIAf8AVJkKDKse)q5Fw`c+nm-peod01c*I6xLy ztb;tn2_jGTc(l`p-N%k6zdpr72iero$J$O{z@wHZs#xb&B#U_VVTNaaeu2Z^@87=o 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I see that this is definitely injective, since the way the exponents + \textit{I'm kinda stuck at this point. + I see that this is definitely injective, since the way the exponents are defined, you will always know which prime factors belong to the numerator or to the denominator. But I fail to prove this using the direct definition of $f$ like we could do for the natural numbers. This is because the products of the denominators in the last equation are not unique. So maybe I simplified them too much and shouldn't try and write them in terms of $x_1$ and $x_2$ like we did earlier, and try and focus more on just the exponents, but I feel it becomes really hard - to show that $x_1 = x_2$ that way. + to show that $x_1 = x_2$ that way.} \item[Surjectivity:] We want to show that $f$ is onto, i.e. $f(\{ q > 0 : q \in \Q \}) = \N$. -- 2.34.1 From ee25ec1b7535125b6ecf63d1a485ac8874ab5dcc Mon Sep 17 00:00:00 2001 From: Philippe Zwietering <philippe.zwietering@gmail.com> Date: Tue, 16 May 2023 15:50:37 +0200 Subject: [PATCH 8/8] Updated main text pdf --- assignments/main_text.pdf | Bin 570002 -> 575515 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/assignments/main_text.pdf b/assignments/main_text.pdf index 84f1e19e6cd4f93a38d416cd81a63857aa81fc52..279e02f5c18e267e77a8d143185f6c2d1cbcedc6 100644 GIT binary patch delta 98956 zcmV(!K;^%Z<szGqBe1qF128o<mtlJdD1Yre*^V5!k?;D7dYf!R&Eb7$G5aupwHO0q 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