Question

Let R be a relation on the set Z of all integers defined as (x, y) ∈ R ⇔ x-y is divisible by 2. Then, the equivalence class [1] is _________________.

Answer 1

Given: R is the equivalence relation on the set Z of integers defined as (x, y) ∈ R ⇔ x − y is divisible by 2.

To find the equivalence class [1], we put y = 1 in the given relation and find all the possible values of x.

Thus,
R = {(x, 1): x − 1 is divisible by 2}
⇒ x − 1 is divisible by 2
⇒ x = ±1, ±3, ±6, ±9, ....

Therefore, equivalence class [0] = {±1, ±3, ±6, ±9, ....}

Hence, the equivalence class [1] is {±1, ±3, ±6, ±9, ....}.