1.

LetA={x inZ : 0lele 12}. showthat R={(a,b):|a-b| isa multipleof 4 is (i)reflexive, (ii)symmetric and (iii)transitive. Findthe setof elementsrelatedto 1.

Answer»

Solution :Clearly,`A={0,1,2,3,4. . .,10,11,12}.`
here,R satisfiesthefollowingproperties .
(i) Reflexivity
Leta be anarbitrary elementof A , then ,
a-a =0, whichis a multipleof 4 .`THEREFORE ` a R a for all ` a in A.`
(ii)SYMMETRY
Leta R b , then
`a R bimplies |a-b|` is a maultipleof 4
`implies |-(a-b)` is a multipleof 4 .
`implies|b-a| ` ismultipleof4
`implies bR a.`
`therefore` R issymmetric .
(iii)tranistivity
Leta R band b Rc . then ,
a R b, b R c
`implies |a-b|` isa multipleof 4and|b-c| ismultipleof 4 .
`Let |a-b|=4k_(1)and |b-c|=4k_(2).` the N ,
`|a-c|=|(a-b)-(b-c)|=|4k_(1)-4k_(2)|`
`=|4(k_(1)-k_(2))|=4|K_(1) -K_(2)|` whichis a multipleof 4
`thereforea R b,b R c implies aRc . so ` R istransitive .
thus,R is reflexive, symmetricand TRANSITIVE .
`Now,[1] ={x in A :x R 1}.`
`={ x in A : |x-1|` isa multipleof 4}
`={1,5,9}.`
HENCE, therequiredsetis `{1,5,9}.`


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