Let R be the relation defined on Real numbers such that R={(a,b)/a-b+root3belongs to S} where S is set of irrational numbers.Is this an equivalence relation.Pls explain fast.Thankyou.

let a belongs R
then a -a + root3 =root3 which belongs to S
Therefore (a,a) belongs R
R is reflexive

Let (a,b) belongs to R for any real numbers a& b
Therefore a-b + root3 is an irrational number
Therefore b-a +root3 will also be an irrational number
so we can say that (b,a) belings to R
So R is symmetric.

Now let (a,b) & (b,c) belongs to R
Therefore a-b +root3 & b-c +root3 are irrational

Adding above two irrational numbers we get a -c +2root3 which is also irrational

(Because if we add two irrational numbers then we get an irrational number)
So if a- c +root3 +root 3 is irrational
If we subtrat root3 from an irrational number we get
a-c +root3 which should also be irrational
Thus (a,c) belongs to R
So R is transitive and thus R is an equivalence relation!

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