# Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given byR = {(a, b) : a = b}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}

For any element a ϵ A, we have (a,a) ϵ R as a = a.

Therefore, R is reflexive.

Now, Let (a,a) ϵ R

a = b

b = a

(b,a) ϵ R

Therefore, R is symmetric.

Now, Let (a,b), (b,c) ϵ R

a = b and b = c

a = c

(a,c) ϵ R

Therefore, R is transitive.

Therefore, R is an equivalence relation.

The set of elements related to 1 will be those elements from set A which are equal to 1.

Therefore, the set of elements related to 1 is {1}.

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