Discrete Mathematics with Applications 4th Edition

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

Chapter 6 - Set Theory - Exercise Set 6.1 - Page 351: 30

Answer

$yes\,\,,\left \{ A_{0},A_{1},A_{2},A_{3} \right \}\,is\,a\,partition\,of\,\,\mathbb{Z} $ $A_{0}=\left \{ n\in \mathbb{Z}\mid n=4k,\,\,for\,some\,integer\,k \right \}$ $A_{1}=\left \{ n\in \mathbb{Z}\mid n=4k+1,\,\,for\,some\,integer\,k \right \} $ $A_{2}=\left \{ n\in \mathbb{Z}\mid n=4k+2,\,\,for\,some\,integer\,k \right \}$ $A_{3}=\left \{ n\in \mathbb{Z}\mid n=4k+3,\,\,for\,some\,integer\,k \right \}$ $\mathbb{Z}\,\,the\,\,set\,of\,all\,integers\,$

Work Step by Step

$according\,\,to\,\,the\,quotient-remainder\,\,theorem$ $any\,integer\,\,n\,\,can\,\,be\,represented\,\,in\,exactly\,\,one\,of\,the\,\,four\,forms\,$ $A_{0}=\left \{ n\in \mathbb{Z}\mid n=4k,\,\,for\,some\,integer\,k \right \}$ $A_{1}=\left \{ n\in \mathbb{Z}\mid n=4k+1,\,\,for\,some\,integer\,k \right \} $ $A_{2}=\left \{ n\in \mathbb{Z}\mid n=4k+2,\,\,for\,some\,integer\,k \right \}$ $A_{3}=\left \{ n\in \mathbb{Z}\mid n=4k+3,\,\,for\,some\,integer\,k \right \}$ $and\,\,this\,implies\,\,that\,\,$ $A_{0}\cup A_{1}\cup A_{2}\cup A_{3}=\mathbb{Z}$ $and\,\,A_{0},A_{1},A_{2},A_{3}\,\,are\,mutually\,\,dis\! joint\,\,(as\,any\,integer\,can\,not\,be\,in\,any\,\,two\,\,of\,the\,sets)$ $so\,\,,\left \{ A_{0},A_{1},A_{2},A_{3} \right \}\,is\,a\,partition\,of\,\,\mathbb{Z} $
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