Algebra Can Be Fun by Yakov Isidorovich Perelman, V. G. Boltyansky, George PDF

By Yakov Isidorovich Perelman, V. G. Boltyansky, George Yankovsky, Sam Sloan

ISBN-10: 0714713538

ISBN-13: 9780714713533

It is a ebook of interesting difficulties that may be solved by using algebra, issues of fascinating plots to excite the readers interest, a laugh tours into the historical past of arithmetic, unforeseen makes use of that algebra is positioned to in daily affairs, and extra. Algebra could be enjoyable has introduced millions of kids into the fold of arithmetic and its wonders. it really is written within the kind of vigorous sketches that debate the multifarious (and exciting!) functions of algebra to the realm approximately us. the following we come upon equations, logarithms, roots, progressions, the traditional and well-known Diophantine research and masses extra. The examples are pictorial, shiny, frequently witty and convey out the essence of the problem to hand. there are various tours into heritage and the background of algebra too. not anyone who has learn this publication will ever regard arithmetic back in a lifeless mild» Reviewers regard it as one of many best examples of renowned technological know-how writing.

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Der anschauli che Begriff "Ankleben einer Zelle " lässt sich mit der kanonischen Projektion p: X U D " --+ X U f D " mathematis ch beschreiben: ple n bildet e" homöomorph auf p(e" ) ab. 26 Beispiele. (a) Sei X := D 2, 1 := ids l. Der Raum X Uf D2 ist eine 2-dim ension ale Sphäre. Vgl. Abb . 4 (a). ° (b) Sei X := {(x , y) E ]R2 I < x ~ 1,0 ~ y < I} , A := {(x ,y) E X I oder I} , Y := [0,1], und I : A --+ Y sei definiert durch I(O,y) = y , 1(1 , y) = 1 - y . Der Raum M := Y Uf X heißt Möbiusband.

Ist f: X -t Y stetig, so gilt: (a) Die Abbildungen p, J und j sind stetig. (b) J ist genau dann ein Homömorphismus, wenn das Bild jeder offenen (abgeschlossenen) Menge der Form f-1(A), Ac Y, offen (abgeschlossen) in f(X) ist. (c) Ist f: X -t Y außerdem surjektiv und offen oder abgeschlossen, so trägt Y die Identijizierungstopologie bezüglich f. Beweis. 3 stetig. Da f(X) die Initialtopologie bezüglich j trägt und f = j 0 J 0 p stetig ist, ist auch J 0 p stetig. Da XI", die Finaltopologie bezüglich p trägt, ist J stetig.

Bezeichnet [x] die größte ganze Zahl , die kleiner oder gleich x ist. Sei, < 1 eine positive irrationale Zahl. Die Abbildung I: Z -+ [0,1[, definiert durch n I-t n,- [n,] ist injektiv, und I(Z) liegt dicht in [0,1[. Wenn Z die diskrete Topologie trägt, ist I natürlich st etig, aber keine Einbettung. 2 (h , i) haben wir auf dem Produkt metrischer Räume (X, , d i ) , i E I , Metriken definiert , die auf den üblichen Einbettungen der Faktorräume in den Produktraum jeweils die ursprünglichen Metriken der Faktorräume eventuell bis auf einen konst anten Faktor genau - induzieren .

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Algebra Can Be Fun by Yakov Isidorovich Perelman, V. G. Boltyansky, George Yankovsky, Sam Sloan


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