find whether the relation, defined on set R, the set of real no. defined as S={ (a,b): a,b belongs to R and a-b+root3 belongs to T, the set of irrational no., is reflexive, symmetric and trasitive. can we say that relation S is not an equivalancerelation?

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Please find below the solution to the asked query:

Our relation is given by:aRb : a-b+3 is irrrational1 Reflexive:aRa: a-a+3=3 which is irrational.Hence aRa is true.R is a reflexive relation.2 SymmetricbRa: b-a+3Now consider a case when a-b=3, thenb-a=-3bRa: -3+3=0 which is rational.Hence bRa does not hold for all a,bR is not a symmetric relation.3 TransitiveLet bRc: b-c+3 is irrationalLet b=23 and c=3 then above statement is true.aRc: a-c+3=a-3+3=a, As aR, hence a can be rational or irrational.aRc: a-c+3 is irrational will not always be true.R is not a transitive relation.As R is neither symmetric nor transitive.Hence we can say that Ris not an equivalence relation.

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