Question

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

Answer 1

⇒ (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| is a multiple of 4.
⇒ (a, c) ∈ R
∴ R is transitive.
Hence, R is an equivalence relation.
The set of elements related to 1 is {1, 5, 9} as

|1 – 1| = 0 is a multiple of 4.
|5 – 1| = 4 is a multiple of 4.
|9 – 1| = 8 is a multiple of 4.
(ii) R = {(a, b): a = b}

∴ 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}.