Answer :

It is given that Relation R in the set Z of all integers defined as

R = {(x, y) : x – y is an integer}


Now, for every x ϵ Z, (x, x) ϵ R, as x –x = 0 is an integer.


R is reflexive.


Now, for every x, y ϵ Z if (x, y) ϵ R, then x – y is an integer.


-(x y) is also an integer.


(y x) is an integer.


(y, x) ϵ R


R is symmetric.


Now, for every (x, y) and (y, z) ϵ R where x, y, z ϵ R,


(x y) and (y z) is an integer.


(x z) = (x y) + (y x) is an integer.


(x, z) ϵ R


R is transitive.


Therefore, R is reflexive, symmetric and transitive.


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