Question

Show 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} is an equivalence relation. Find the set of all elements related to 1.

Answer 1

A = {x ∈ Z : 0 ≤ x ≤ 12} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

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

(i) Reflexive:

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

∴ R is reflexive.

(ii) Symmetric:

Now, let (a, b) ∈ R

⇒ |a – b| is a multiple of 4.

⇒ |–(a – b)| = |b – a| is a multiple of 4.

⇒ (b, a) ∈ R

Thus (a, b) ∈ R 

⇒ (b, a) ∈ R

∴ R is symmetric.

(iii) Transitive:

Now, let (a, b), (b, c) ∈ R.

⇒ |a – b| is a multiple of 4 and |b – c| is a multiple of 4.

⇒ |a – c| = |a – b + b – c| = |a – b| + |b – c|

⇒ (a – c) = (a – b) + (b – c) is a multiple of 4.

⇒ (a, c) ∈ R  ...[∴ |a – b| is multiple of 4 and |b – c| is multiple of 4.]

∴ R is transitive.

Hence, R is an equivalence relation.

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

|1 – 1| = 0 is a multiple of 4.

|5 – 1| = 4 is a multiple of 4.

|9 – 1| = 8 is a multiple of 4.