Discrete Mathematics with Applications 4th Edition

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

Chapter 8 - Relations - Exercise Set 8.2 - Page 458: 14

Answer

Not Reflexive Symmetric Not Transitive

Work Step by Step

Reflexive : for all x $\in$ Z , xOx <=> x-x is odd Counter Example : take any int lets say 2, 2-2 = 0 zā‰  2k+1 (hence (x,x) $\notin$ O) Symmetric : for all x,y $\in$ Z , if xOy then yOx xOy <=> x-y = 2k+1 -x+y = -2k - 1 (Multiply both sides by -1) y- x = 2{-k} - 1 (Commutive Law) y-x = odd (note that -k $\in$ Z hence y-x is odd) yOx (by Def of O) Transitive: for all x,y and z $\in$ Z if (xOy and yOz) then ( xOz) Counter Example: take x= 1, y=2 ,z=1 xOy = -1 #odd yOz = 1 #odd but xOz = 0 # even , hence the statement is false, hence not transitive)
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