Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence and characterize its equivalence classes

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence
and characterize its equivalence classes
Transcribed Image Text:Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence and characterize its equivalence classes
Expert Solution
Step 1
Given that the relation < over real x < y if only if (x – y)ez.
Test for reflexive:
x<x VIER since x- x= 0 eZ.
Hence, x<x and < is reflexive.
Test for Symmetric:
If x< y then y< x for every x,yeR.
x< y= x – y E Z
—- (у-х)€z
— у -хеZ
Vx, yER
= y<x
Hence, < is symmetry.
Step 2
Test for Transitive:
If x< y then y<z then x< z
x< y=x- yE Z
y<x= y- xEZ
x- y + y – z =x-zEZ
Hence, < is transitive.
Thus, it is an equivalence relation.
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