Answer :

It is given that the relation R in the set A = {x Z : 0 x 12}, given by

R = {(a, b) : |a – b| is a multiple of 4}


For any element a ϵ A, we have (a,a) ϵ R as |a-a|=0 is a multiple of 4.


Therefore, R is reflexive.


Now, Let (a,a) ϵ R


|a b| is a multiple of 4


|b a| = |a b| is a multiple of 4


(b,a) ϵ R


Therefore, R is symmetric.


Now, Let (a,b), (b,c) ϵ R


|a b| is a multiple of 4 and |b - c| is a multiple of 4


|a c| = |(a b) + (b - c)| is a multiple of 4


(a,c) ϵ R


Therefore, R is transitive.


Therefore, R is an equivalence relation.


The set of elements related to 1 is {1,5,9}


|1-1| = 0 is multiple of 4


|5-1| = 4 is multiple of 4


|9-1| = 8 is multiple of 4.


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