Question

Check whether the relation R in the set z of integers defined as R = {(a, b) : a+b is divisible by 2is an equivalence relation. Write the equivalence class [0].

Answer 1

a R b⇔a−b divisible by 3

Reflexive:

a R a⇔a−a divisible by 3 True

Therefore, the given relation R is a reflexive relation.

Symmetric:

a R b⇔a−b divisible by 3⇒ a−b=3k, where k is an integer.

Then,

b R a⇔b−a divisible by 3

Because b−a=−(a−b)=−3k which is divisible by 3.

Therefore, the given relation R is a symmetric relation.

Transitive:

a R b⇔a−b divisible by 3⇒ a−b=3k, where k is an integer.

b R c⇔b−c divisible by 3⇒ b−c=3p, where, p is an integer

Then,

a R c⇔a−c divisible by 3

Because a−c=(a−b)+(b−c)=3k+3p=3(k+p) which is divisible by 3.

Therefore, the given relation R is a transitive relation.

Since, the given relation R satisfies the reflexive, symmetric, and transitive relation properties, Therefore, it is an equivalence relation.