Let R 1 = a , b ∈ N × N : a - b ≤ 13 and R 2 = a , b ∈ N × N : a - b ≠ 13…

To check if $R_1$ is transitive -

Let $a=16, b=4, c=1$

Here $\left|a-b\right|=\left|16-4\right|\le 13$ holds true

now $\left|b-c\right|=\left|4-1\right|\le 13$ $\Rightarrow \left|3\right|⩽13$   also holds true

but $\left|a-c\right|=\left|16-1\right|\nleqq 13$

Hence $R_1$ is not transitive

Similarly, to check transitivity of $R_2$

Let $a=16, b=4, c=3$

Here $\left|a-b\right|=\left|16-4\right|\ne 13 \text{and} \left|b-c\right|=\left|4-3\right|\ne 13$

but $\left|a-c\right|=\left|16-3\right|=13$

Hence, $R_2$ is also not transitive.

Hence neither $R_1$ nor $R_2$ is an equivalence relation