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zsb-+iOcC&Zlj=`JVxEwPp?R7)GN_jNO-12UnZY2_n<5tPGCGZ&T(cCn?F2b>`sL2F rMo?+03FdE>1e)By3tJ&U|F?W)UK_)GMCIV&XXoZXqM?yil|lLs^VBV= diff --git a/assignments/week2/main.tex b/assignments/week2/main.tex index 62a1eac..c4a2bd2 100644 --- a/assignments/week2/main.tex +++ b/assignments/week2/main.tex @@ -4,6 +4,19 @@ \section{Week 2} \exercise*[1.1.1] -Exercise 1.1.1 comes here. +\begin{tcolorbox} + Prove: + + Let $F$ be an ordered field and $x,y,z \in F$. If $x < 0$ and $y < z$, then $xy > xz$. +\end{tcolorbox} + +So let's assume the premise. $F$ is an ordered field and $x,y,z \in F$, +and we choose $x,y$ and $z$ such that $x < 0$ and $y < z$. + +From $x < 0$ it follows that $(-x) > 0$. From $y < z$ it follows that $0 < z - y$. +From both of these, we can conclude that $0 < (-x)(z-y)$. +Working out the right side with the distributive law, gives $0 < (-x*z)-(-x*y)$. +Using $-1*-1 = 1$, gives $0 < (-xz)-(-xy)$, thus $0 < xy - xz$. The right part can be split again: $xz < xy$. +Then, the < can be flipped, which gives $xy > xz$. \qed \end{document}