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 7.1 - Page 396: 46

Answer

First Case: Assume $\ f^-1$(C U D) is true by def of range f(y) = x , x$\in$(C U D) **( x $\in$ C or x $\in$ D by def of union) $\ f^-1$(y) = x , x$\in$ C implies x$\in$$\ f^-1$(C) or $\ f^-1$(y) = x , x$\in$ D implies x$\in$$\ f^-1$(D) by def of union x $\in$ $\ f^-1$(C) U $\ f^-1$(D) Second Case : Assume $\ f^-1$(C) U x$\in$$\ f^-1$(D) is true hence, x $\in$ $\ f^-1$(C) U $\ f^-1$(D) x$\in$ $\ f^-1$(C) $OR $ x$\in$$\ f^-1$(D) **By definition of union which implies $\ f^-1$(y) = x , x $\in$ (C U D) which also means x $\in$ $\ f^-1$(C U D) In conclusion, we got from : Case 1: $\ f^-1$(C U D) $\subseteq$ $\ f^-1$(C) U $\ f^-1$(D) Case 2: $\ f^-1$(C) U $\ f^-1$(D) $\subseteq$ $\ f^-1$(C U D) hence, $\ f^-1$(C U D) $=$ $\ f^-1$(C) U $\ f^-1$(D)

Work Step by Step

First Case: Assume $\ f^-1$(C U D) is true by def of range f(y) = x , x$\in$(C U D) **( x $\in$ C or x $\in$ D by def of union) $\ f^-1$(y) = x , x$\in$ C implies x$\in$$\ f^-1$(C) or $\ f^-1$(y) = x , x$\in$ D implies x$\in$$\ f^-1$(D) by def of union x $\in$ $\ f^-1$(C) U $\ f^-1$(D) Second Case : Assume $\ f^-1$(C) U x$\in$$\ f^-1$(D) is true hence, x $\in$ $\ f^-1$(C) U $\ f^-1$(D) x$\in$ $\ f^-1$(C) $OR $ x$\in$$\ f^-1$(D) **By definition of union which implies $\ f^-1$(y) = x , x $\in$ (C U D) which also means x $\in$ $\ f^-1$(C U D) In conclusion, we got from : Case 1: $\ f^-1$(C U D) $\subseteq$ $\ f^-1$(C) U $\ f^-1$(D) Case 2: $\ f^-1$(C) U $\ f^-1$(D) $\subseteq$ $\ f^-1$(C U D) hence, $\ f^-1$(C U D) $=$ $\ f^-1$(C) U $\ f^-1$(D)
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