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 8.3 - Page 476: 25

Answer

A is the “absolute value” relation on R: For all real numbers x and y,x A y ⇔ |x| = |y|. Reflexive: for all x ∈ Z xAx true since |x| = |x| for every x Symmetric: for all x,y ∈ Z if xAy then yAx xAy <=> |x| = |y| >> |y| = |x| (commutive law) = yAx (by def of A) Transitive: for all x,y, and z ∈ Z if (xAy and yAz) then xAz xAy <=> |x| = |y| >> |y| = |x| commutive law--(1) yAz <=> |y| = |z| -- (2) **sub 1 in 2 |x| = |z| xAz (By definition of A) [that's what we needed to show] Since A is reflective symmetric and transitive, it's an equivalence relation x = 0 , then 0 is the only value with absolute value = 0 or x doesn't equal 0, then {x,-x} are the value with their absluoate = x hence the equvillance classes are {0},{x,-x} for all x $\in$ R

Work Step by Step

A is the “absolute value” relation on R: For all real numbers x and y,x A y ⇔ |x| = |y|. Reflexive: for all x ∈ Z xAx true since |x| = |x| for every x Symmetric: for all x,y ∈ Z if xAy then yAx xAy <=> |x| = |y| >> |y| = |x| (commutive law) = yAx (by def of A) Transitive: for all x,y, and z ∈ Z if (xAy and yAz) then xAz xAy <=> |x| = |y| >> |y| = |x| commutive law--(1) yAz <=> |y| = |z| -- (2) **sub 1 in 2 |x| = |z| xAz (By definition of A) [that's what we needed to show] Since A is reflective symmetric and transitive, it's an equivalence relation x = 0 , then 0 is the only value with absolute value = 0 or x doesn't equal 0, then {x,-x} are the value with their absluoate = x hence the equvillance classes are {0},{x,-x} for all x $\in$ R
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