Question

Let R be the equivalence relation on the set Z of integers given by R = {(a, b): 3 divides a-b}. Then the equivalence class [0] is equal to ____________________.

Answer 1

Given: R is the equivalence relation on the set Z of integers given by R = {(a, b): 3 divides a − b}.

To find the equivalence class [0], we put b = 0 in the given relation and find all the possible values of a.

Thus,
R = {(a, 0): 3 divides a − 0}
⇒ a − 0 is a multiple of 3
⇒ a is a multiple of 3
⇒ a = 3n , where n ∈ Z
⇒ a = 0, ±3, ±6, ±9, ....

Therefore, equivalence class [0] = {0, ±3, ±6, ±9, ....}

Hence, the equivalence class [0] is equal to {0, ±3, ±6, ±9, ....}.