# Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given byR = {(a, b) : |a – b| is a multiple of 4}is an equivalence relation. Find the set of all elements related to 1 in each case.

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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