By Catherine Meusburger
Read Online or Download Lineare Algebra 1 und 2 [Lecture notes] PDF
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Additional resources for Lineare Algebra 1 und 2 [Lecture notes]
Sample text
Dann können wir zu jedem Element y ∈ N ein Urbild xy ∈ M mit f (xy ) = y wählen und eine Abbildung g : N → M , y → xy definieren. Es folgt (f ◦ g)(y) = f (xy ) = y für alle y ∈ N und somit f ◦ g = idN . ⇐: Seien f : M → N und g : N → M Abbildungen mit f ◦ g = idN . Dann gilt (f ◦ g)(y) = f (g(y)) = y für alle y ∈ N . Also existiert zu jedem y ∈ N ein Element x ∈ M mit f (x) = y, nämlich x = g(y), und somit ist f surjektiv. 42 2. ⇒: Sei f : M → N injektiv. Für y ∈ N gilt entweder f −1 ({y}) = ∅ oder es existiert genau ein xy ∈ M mit f (xy ) = y.
4. Komplementsgesetz: Für jede Teilmenge D ⊆ A gilt A \ (A \ D) = D. Beweis: 1. Die Kommutativgesetze ergeben sich direkt aus der Definition von A ∩ B und A ∪ B und der Kommutativität der elementaren Verknüpfungen ∨ und ∧: A ∪ B = {a|(a ∈ A) ∨ (a ∈ B)} = {a|(a ∈ B) ∨ (a ∈ A)} = B ∪ A A ∩ B = {a|(a ∈ A) ∧ (a ∈ B)} = {a|(a ∈ B) ∧ (a ∈ A)} = B ∩ A. 7, 1. reicht es zum Beweis der restlichen Aussagen, jeweils zu zeigen, dass die Menge auf der rechten Seite der Gleichung eine Teilmenge der Menge auf der linken Seite ist und umgekehrt.
Mit vollständiger Induktion kann man beweisen, dass für eine assoziative Verknüpfung ◦ auf einer Menge M alle möglichen Klammerungen eines Ausdrucks a1 ◦ a2 ◦ . . ◦ an das gleiche Element von M liefern. Aus diesen Gründen darf man die Klammern weglassen und stattdessen a1 ◦ ... ◦ an schreiben. 3. Eine Verknüpfung ◦ auf einer Menge M besteht immer aus zwei Dingen: einer Verknüpfungsvorschrift, die angibt, wie man aus zwei gegebenen Elementen a, b ∈ M das Element a ◦ b berechnet, und der Menge M selbst.