From 67ef0d570ef0a52ef9bf5ec1013268cf32a89908 Mon Sep 17 00:00:00 2001 From: raj_mathe Date: Sat, 21 Nov 2020 15:25:26 +0100 Subject: [PATCH] =?UTF-8?q?master=20>=20master:=20k=C3=BCrzer=20Gleichunge?= =?UTF-8?q?n?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- docs/loesungen.pdf | Bin 397355 -> 397357 bytes docs/loesungen.tex | 39 ++++++++++++++++++--------------------- 2 files changed, 18 insertions(+), 21 deletions(-) diff --git a/docs/loesungen.pdf b/docs/loesungen.pdf index 0bdd56a5ac9867ec20682e8efc211b41c63b714a..6d2166d24420943c5153f34aba34aa51331c3238 100644 GIT binary patch delta 25260 zcmV)MK)AoFpct*77=W|^Y7hc9Ig?-}Dt}yAj~h1-(AArEC@<@@6&BvT*_e@nW`Kv1gj4aEQ<5MCfHyVJixIM& zl2F|_g=Vppo6`#C)SJ8{k{jsSHGi1BJek7WCa0lc(j^1obs3wkH}L6=nBo&=K5eI- z1&A>ar#uWG3*%ce3Q%aa6F|BT&{-BBNhsCB0QE*p0SYr80Z1^StUd@I!40vLkKq%W z?T63lR(u4ZObz1GA2G#e8m2y`L9!&ulm%DBD)YG}rooC)gSaNb+CdZZ&3|I69W^M2 zj3QibQJmgDDKKHe({{#LTr^Qy9R`aniB@bBme6b`SQ^ih^*FH7W}7{w1EsL|`Hr9> z6nG_b5Gq1II8b5?mDp@QRL=6K2*!r3s5emGiu{bnw1pE{x(F;)`MIH25W`^wbZH1R zyv0Q(q1U&Hjoh4GZie2pRew4sFKZ*JpU*od7ic}FCFqbOOj5xM5EmpvN5X8uO6XN< zTP&LPlvu$^z@I23VWes2!oP{1oAz@L0w=1ZXy*XSHAFJ}5WF2ni8%}rAzQTXrp?QX z;!RN`qGMA7evny-Qaj*b_R5G>+_)OkWO+iXqe#`kQI7Ai!&`3=kpv^<6h@_HjL3N|)aioL*XC&j!D*r4dFJT0w@2$qec;xcQPApkJ0KrXYOnXn_8^GJeO87bm*ny0`)N+%Po~DVkg9-oEMZ%O~5o6k^VmJb0MTfwE^UwzJzJEYnemUWGQq}bd@Io

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Buchstabe in $w$}\\ - \end{mathe} + wobei $f:W\to\Sigma$ die Abbildung mit + $f(w)=\text{1. Buchstabe in $w$}$ + für alle $w\in W$ ist. Dann per Konstruktion \uline{reduziert} $f$ die Relation $(W,\sim)$ auf $(\Sigma,=)$. @@ -3712,13 +3709,13 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. \textbf{Zu zeigen:} $|\prod_{i=1}^{n}E_{i}|=\prod_{i=1}^{n}|E_{i}|$ gilt.\\ Es gilt - \begin{mathe}[mc]{rcl} + \begin{mathe}[mc]{rclql} |\prod_{i=1}^{n}E_{i}| &= &|\prod_{i=1}^{n-1}E_{i}\times E_{n}|\\ - &= &|\prod_{i=1}^{n-1}E_{i}|\cdot|E_{n}|\\ - &&\text{da $\Phi(2)$ gilt}\\ - &= &\prod_{i=1}^{n-1}|E_{i}|\cdot|E_{n}|\\ - &&\text{wegen der IV}\\ + &= &|\prod_{i=1}^{n-1}E_{i}|\cdot|E_{n}|, + &\text{da $\Phi(2)$ gilt}\\ + &= &\prod_{i=1}^{n-1}|E_{i}|\cdot|E_{n}| + &\text{wegen der IV}\\ &= &\prod_{i=1}^{n}|E_{i}|.\\ \end{mathe} @@ -3781,17 +3778,17 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion. sind $X\times Y'$ und $X\times\{y_{0}\}$ ebenfalls disjunkt. Es folgt - \begin{longmathe}[mc]{RCL} + \begin{longmathe}[mc]{RCLqL} |X\times Y| &= &|X\times (Y'\cup\{y_{0}\}|\\ &= &|(X\times Y')\cup (X\times\{y_{0}\})|\\ - &= &|X\times Y'| + |X\times\{y_{0}\}|\\ - &&\text{wegen Disjunktheit}\\ - &= &|X|\cdot(n-1) + |X|\cdot 1\\ - &&\text{wegen des Falls für $1$-elementigen Mengen}\\ - &= &|X|\cdot n\\ - &&\text{wegen der rekursiven Definition von Multiplikation}\\ - &= &|X|\cdot |Y|,\\ + &= &|X\times Y'| + |X\times\{y_{0}\}| + &\text{wegen Disjunktheit}\\ + &= &|X|\cdot(n-1) + |X|\cdot 1 + &\text{wegen Fall für $1$-elem. Mengen}\\ + &= &|X|\cdot n + &\text{wegen rekursiver Defn von Multiplikation}\\ + &= &|X|\cdot |Y|.\\ \end{longmathe} \end{kompaktenum}