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z{>B;Sb(7TqzCmspHv^^Ab2og`SaqRl&JC-@E4I^K%Hs7FTv#>--ur;`P9mkj%!9?& zVg_dOmArMKGRw$?Cn{b|BBf7ukn8$raD6QF$pUY;cH^p-8 zitvQf2y0t>vF?6(vsK>x_hkAlHU;ZMkf-$8@zNzpiTnA%i+*$8CQ8EJ{YaXsIa7~5 zv!mwcP%9mo>yjwV8moXlx1$X!x4zrY)$jjeub@LG7hEsQOoT<5>2-YZ}RKWV?ckh8vgR+O9 z?(ek-^I$|``6C&DG(+2s#Y-;0DmUph@`ZO`0H79GT81^{FHUHPUaq*79g8im-!imv z4|jvJDi8q`bBFPa(e?edvOsX+?zs>Tso@_q{F=&iCK(feG;=}$fDL42HGpSUxACMGHafz{uaY(UqNbqt>NeGet|1C(L2mS9ofsKndla&oX2PB#sXM!Yuu_r3B4!mzJc#}9Cbl-82osJZ#ivFD}gvfIOP2@UPp zO=1%&ZU8sxKS~5Hp&u5dbMK3Pl$bD~7~_yJ1o98$(;`R^stAR9 zxr{Mz35JYA9FL3aD-f%EfU8-Ap~M)l{)4F17{#{;z^55fO|%I6J1HD(tipWz!iEHK z02NW3c*F#Y+8CPJgeaySPQ4y+p&o|49;%5xT$|ro>HcGIxr>i^rWKn*&A+jwe0wId zCdIcG#nMNYt-YG{#WE^@0F)SUy`1#cocXwi_-7A_X%D`34@v5MG2bUyc})0>nJhW8 VPYFPT$i>Ue!var5C8;b0{~y 0 : q \in \Q \}$, $f(x_1) = f(x_2)$. + 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}

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