Question

Show that 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.

Answer 1

R = {(a, b) : a = b}

(i) Reflexive:

R = {(0, 0), (1, 1), ..., (12, 12)} 

A = {0, 1, 2, ..., 12}

For any element a ∈ A, we have (a, a) ∈ R, since a = a.

∴ R is reflexive.

(ii) Symmetric:

Now, let (a, b) ∈ R.

⇒ a = b

⇒ b = a

⇒ (b, a) ∈ R

∴ R is symmetric.

(iii) Transitive:

Now, let (a, b) ∈ R and (b, c) ∈ R.

⇒ a = b and b = c

⇒ a = c

⇒ (a, c) ∈ R

∴ R is transitive.

Hence, R is an equivalence relation.

The elements in R that are related to 1 will be those elements from set A which are equal to 1.

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