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 6.2 - Page 365: 12

Answer

$for\,\,all\,sets\,\,A\,\,, A\cup \varnothing =A \\ to\,\,prove\,\,this\,\,we\,must\,prove\,\,that\,\,\\ 1-A\cup \varnothing \subseteq A \\ 2-A \subseteq A\cup \varnothing \\ proof\,\,of\,\,1:\\ x\in A\cup \varnothing \overset{def.of union}{\rightarrow} x\in A\,\,or\,\,x\in\varnothing \\ since\,\,\varnothing\,\,is\,the\,empty\,set\rightarrow x\,must\,belong\,to\,A\rightarrow x\in A\\ so\,\,A\cup \varnothing \subseteq A \\ proof\,\,of\,\,2:\\ x\in A \overset{def.of\,union}{\rightarrow}x\in A\cup \varnothing\\ so\,\,A \subseteq A\cup \varnothing \\ from\,1\,,2\, \,\,, A\cup \varnothing =A$

Work Step by Step

$for\,\,all\,sets\,\,A\,\,, A\cup \varnothing =A \\ to\,\,prove\,\,this\,\,we\,must\,prove\,\,that\,\,\\ 1-A\cup \varnothing \subseteq A \\ 2-A \subseteq A\cup \varnothing \\ proof\,\,of\,\,1:\\ x\in A\cup \varnothing \overset{def.of union}{\rightarrow} x\in A\,\,or\,\,x\in\varnothing \\ since\,\,\varnothing\,\,is\,the\,empty\,set\rightarrow x\,must\,belong\,to\,A\rightarrow x\in A\\ so\,\,A\cup \varnothing \subseteq A \\ proof\,\,of\,\,2:\\ x\in A \overset{def.of\,union}{\rightarrow}x\in A\cup \varnothing\\ so\,\,A \subseteq A\cup \varnothing \\ from\,1\,,2\, \,\,, A\cup \varnothing =A$
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