Let the relation be R on Z is given by R={(a,b):5dividesa−b}.
We observe the following properties of relation R
Reflexivity: for any a∈Z
a−a=0=0×5
⇒ 5 divides a−a⇒(a,a)∈R
So, R is relexive relation on Z
Symmetry: Let a,b∈Z be such that
(a,b)∈R
⇒ 5 divides a−b
⇒ a−b=5λ for some λ∈Z
⇒ b−a=5(−λ), where −λ∈Z
⇒ 5 divides b−a⇒(b,a)∈R
Thus (a,b)∈R⇒(b,a)∈R. So, R is a symmetric relation on Z.
Transitivity. Let a,b,c∈Z be such that (a,b)∈R and (b,c)∈R. Then,
(a,b)∈R⇒5 divides a−b⇒a−b=5λ for some λ∈Z
and (b,c)∈R⇒5 divides b−c⇒b−c=5μ for some μ∈Z
∴ a−b+b−c=5(λ+μ)
⇒ a−c=5(λ+μ), where λ+μ∈Z
⇒ 5 divides a−c
⇒ (a,c)∈R
thus, (a,b)∈R and (b,c)∈R⇒(a,c)∈R
So, R is a transitive relation on Z
Hence, R is an equivalence relation on Z.