diff --git a/assignments/week1/1.pdf b/assignments/week1/1.pdf index 33735dc..257c441 100644 Binary files a/assignments/week1/1.pdf and b/assignments/week1/1.pdf differ diff --git a/assignments/week1/1.tex b/assignments/week1/1.tex index 05de978..3cada10 100644 --- a/assignments/week1/1.tex +++ b/assignments/week1/1.tex @@ -204,13 +204,14 @@ 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} - I'm kinda stuck at this point. 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. $f(\{ q > 0 : q \in \Q \}) = \N$.