Answer :

In order to show R is an equivalence relation we need to show R is Reflexive, Symmetric and Transitive.


Given that, a, b S, R = {(a, b) : a = ± b }


Now,


R is Reflexive if (a,a) R a S


For any a S, we have


a = ±a


(a,a) R


Thus, R is reflexive.


R is Symmetric if (a,b) R (b,a) R a,b S


(a,b) R


a = ± b


b = ± a


(b,a) R


Thus, R is symmetric .


R is Transitive if (a,b) R and (b,c) R (a,c) R a,b,c S


Let (a,b) R and (b,c) R a, b,c S


a = ± b and b = ± c


a = ± c


(a, c) R


Thus, R is transitive.


Hence, R is an equivalence relation.


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