From 16ca358a0823e7340e5ce566a7adbe93cd2682e0 Mon Sep 17 00:00:00 2001 From: Ameyah Date: Sat, 6 Apr 2019 17:47:41 +0200 Subject: [PATCH] fixed typos and errors in LAAG1+2! --- .../TeX_files/Abbildungen.tex | 29 ++++++++++-------- .../LAAG 1+2 Semester_Fehm/Vorlesung LAAG.pdf | Bin 1251846 -> 1620070 bytes 2 files changed, 17 insertions(+), 12 deletions(-) diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Abbildungen.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Abbildungen.tex index 016f670..dcd9dd1 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Abbildungen.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Abbildungen.tex @@ -50,15 +50,14 @@ \end{itemize} \end{example} -\begin{example}[Eigenschaften von Funktionen] - \begin{itemize} - \item \begriff{injektiv}: Zuordnung ist eindeutig: $F(m_1) = F(m_2) \Rightarrow m_1=m_2$ \\ - Bsp: $x^2$ ist nicht injektiv, da $F(-2)=F(2)=4$ - \item \begriff{surjektiv}: $F(M)=N$ ($\forall n \in N \; \exists m \in M \mid F(m)=n$) \\ - Bsp: $\sin(x)$ ist nicht surjektiv, da es kein $x$ für $y=27$ gibt - \item \begriff{bijektiv}: injektiv und surjektiv - \end{itemize} -\end{example} +\begin{definition}[Eigenschaften von Abbildungen] + Sei $f: X \to Y$ eine Abbildung. + \begin{enumerate} + \item $f$ ist \begriff{injektiv},wenn $\forall x,y \in X (f(x) = f(y) \Rightarrow x=y)$. + \item $f$ ist \begriff{surjektiv}, wenn $\forall y \in Y \exists x \in X (f(x)=y)$ ($\forall n \in N \exists m \in M \mid f(m)=n$). + \item $f$ ist \begriff{bijektiv}, wenn $f$ injektiv und surjektiv ist. + \end{enumerate} +\end{definition} \begin{example} \begin{itemize} @@ -72,14 +71,20 @@ \begin{definition}[Einschränkung] \proplbl{1_2_6} Sei $f: x \mapsto y$ eine Abbildung. Für $A \subset X$ - definiert man die \begriff{Einschränkung}/Restrikton von $f$ auf $A$ als die Abbildung + definiert man die \begriff{Einschränkung}/\begriff{Restrikton} von $f$ auf $A$ als die Abbildung \begin{align} f \vert_A:\begin{cases} A \to Y \\ a \mapsto f(a) \end{cases}\notag \end{align} - Das \begriff{Bild} von $A$ unter $f$ ist $f(A) := \{f(a): a \in A\}$. \\ - Das \begriff{Urbild} einer Menge $B \subset Y$ unter $f$ ist $f^{-1} := \{x \in X: f(x) \in B\}$. \\ + Das \begriff{Bild} von $A$ unter $f$ ist + \begin{align} + f(A) := \{f(a): a \in A\}. \notag + \end{align} + Das \begriff{Urbild} einer Menge $B \subset Y$ unter $f$ ist + \begin{align} + f^{-1} := \{x \in X: f(x) \in B\}. \notag + \end{align} Man nennt $\Image(f) := f(X)$ das Bild von $f$. \end{definition} diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/Vorlesung LAAG.pdf b/2. 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