Question

Let $R$ be the equivalence relation on the set $Z$ of integers given by $R=\{(a,b):2\text{divides}a-b\}$. Write the equivalence class {0}.

Answer 1

R = (a,b) : 2 divides (a-b)

$\Rightarrow (a-b)$ is a multiple of 2.

To find equivalence class $0,$ put $b=0$

So, $(a-0)$ is a multiple of 2

$\Rightarrow$ a is a multiple of 2

So, In set z of integers, all the multiple

of 2 will come in equivalence

class $\left\{0\right\}$

Hence, equivalence class $\left\{0\right\}=\left\{2x\right\}$

where x = integer (z)