Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 7 - Functions - Exercise Set 7.2 - Page 414: 12

Answer

a- (i)one-to-one (ii) not onto b- onto

Work Step by Step

$a-\\ f:\mathbb{Z}\rightarrow \mathbb{Z},f(n)=2-3n\\ A\,function\,\,f: \mathbb{Z} \rightarrow \mathbb{Z}\,is\,\,one-to-one\,\Leftrightarrow \\ \forall \,\,x_{1}\,and\,x_{2}\,\,in\,\,\mathbb{Z}\,\,if\, f(x_{1}) = f(x_{2})\,\,then\,x_{1} = x_{2}.\\ so\,let\,f(x_{1})=f(x_{2})\\ \therefore 2-3x_{1}=2-3x_{2}\Leftrightarrow -3x_{1}=-3x_{2}\Leftrightarrow x_{1}=x_{2}\\ \therefore f\,is\,one-to-one\\ f\,is\,not\,onto\,as\,7\in \mathbb{Z}\,\\ is\,not\,image\,of\,some\,element\,in\,\mathbb{Z}\\ (because\,2-3n=7\Leftrightarrow -3n=5\Leftrightarrow n=\frac{-5}{3}\notin \mathbb{Z})\\ $ $b-\\ G:\mathbb{R}\rightarrow \mathbb{R},G(x)=2-3x\\ G: \mathbb{R} \rightarrow \mathbb{R} \,\,is\,\,onto\,\Leftrightarrow \,\\ \forall x\,in\,\mathbb{R} ,\exists y \in \mathbb{R}\,such\,that\, G(x) = y.\\ so\,let\,y\in \mathbb{R}\\ define\,\,x=\frac{2-y}{3}\\ by\,def.\,of\,G(x)\\ G(x)=2-3(\frac{2-y}{3})=2-2+y=y\\ so\,for\,any\,y\in \mathbb{R}\\ y\,is\,image\,of\,some\,element\,in\,\mathbb{R} $
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