From ad4aaab3ebabd5d8f5680a8b59065920a3e35f38 Mon Sep 17 00:00:00 2001 From: Philippe Zwietering Date: Tue, 16 May 2023 15:35:53 +0200 Subject: [PATCH] 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 zf9|=d$MF6Y@;Le^?f=$pU@LqNI?fTo#y_piQ^%#@ea~?UDeF1OZ zIc?dPs!(69(nzdTx&1s(MB<1+So3q|L-=g(<4}*aaI`wg!*Q{-0eAB8F$uUK?F^EV ztFHf?9|L^l0Zy)}=HxI1Gia7V{crg-9I8Q~Oi6{#7EFHzhdJh?uvrfyop-}m!c(CF 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zT)F*Q{CY4yk)5Tm6chVLkLn0jW(*~^jw>k@&;bRB<u-t+76>ik1Zz+c3eXEgR0n{Z zDj~81U@Vp3-Ib6L_d+DjG{^)VtEjBPvDW@;>}VCH&lM<Y72<IgNDd=JeKiEjH4iUL z*%yhs1U`lmWPGNI_QBEqjS@y0DNKL{fFzemHZ0?VGX)??OASDj7^7Jfg19yyJ~lwY zD}~bA^^+YL!*UpFOShSw)ctyXGX7iDBtR^$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}