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On theset Z of allintegers , consider the relation R={(a,b):(a-b) is divisible by 3}. Show thatR isanequivalencerelationon Z. Also findthepartitioning of Z intomutuallydisjointequivalenceclasses . |
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Answer» Solution :the relationR on Z satisfiesthe followingproperties : (i) Reflexivity Let ` a in Z` then,`(a-a)=0,` whichisdivisibleby 3 . `thereforea Ra AAa in Z.` So,R isreflexive .(ii)SYMMETRY Let` a ,bin Z` suchthat`a ,R,b` then `a R bimplies(a-b) ` isdivisibleby 3 `implies -(a-b)` isdivisibleby 3 `implies (b-a) ` is divisible by 3 `impliesb R a .` `thereforea R b impliesbRa AAa,bin Z .` So,R issymmetric . (iii)Transitivity Leta,b,c `in Z ` such thata R bandb R c. then `a R b,bRc implies (a-b)` isdivisibleby3 and(b-c)isdivisibleby 3 `implies [(a-b)+(b-c)] `isdivisibleby 3 `implies(a-c)` is DIVISIBLEBY 3. thus,`a R b ,B R cimplies aR cAA a,b,cin Z.` ` therefore ` R isequivalencerelation on Z. Now ,letusconsider [0],[1] and [2] we have : `[0]={x in Z : x R O }` ` ={x in Z: (x-0)` is divisibleby 3} `={. . . . . .,-6,-3,0,3,6,9,. . . . }.` `therefore [0] {. . . .,-6,-3,0,3,6,9,....}.` Similarly ,`[1] ={x INZ: xR 1}` `={x inz : (x-1) ` is divisibleby 3} `={. . . . .,-5,-2,1,4,7,10,. . . . . .,}.` and,`[2] ={x inZ: x R 2}.` `={x in Z : (x-2)` is divisibleby3} `={. . . ,-4,-1,2,5,8,11,. . .}.` ` therefore [2] ={. .. ..,-4,-1,2,5,8,11,. . . .}.` CLEARLY[0] ,[1]and [2]aremutuallydisjoint and`Z=[0] cup[1]cup [2].` |
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