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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. |
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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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