R is a relation on the set Z of integers and it is given by (x, y) ∈R…

The correct answer is Option (2) → reflexive and symmetric

For any $x ∈ Z$, we have

$|x-x|=0≤1$

$∴|x-x|≤1$   for all $x ∈ Z$

$⇒(x, x) ∈ R$   for all $x ∈ Z$

⇒ R is reflexive on Z.

Let $(x, y) ∈ R$. Then,

$|x-y|≤1⇒|y-x|≤1⇒ (y, x) ∈ R$

Thus, $(x, y) ∈ R⇒(y, x) ∈ R$

So, R is a symmetric relation on Z.

We observe that $(1, 0) ∈ R$ and $(0, -1) ∈ R$, but $(1, -1) ∉ R$.

So, R is not a transitive relation on Z.