Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by

R = {(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}.