Let R _ 1 = \ (a, b) ∈ N × N: |a - b| ≤ 13\ and R _ 2 = \ (a, b) ∈...

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$holds true</p><p>now \left|b - c\right| = \left|4 - 1\right| \le 13 ⇒ \left|3\right| ⩽ 13$ also holds true

but \left|a - c\right| = \left|16 - 1\right| ≰ 13$</p><p>Hence${R}_{1}$is not transitive</p><p>Similarly, to check transitivity of${R}_{2}$</p><p>Let$a = 16, b = 4, c = 3$</p><p>Here \left|a - b\right| = \left|16 - 4\right| \ne 13\text{and}\left|b - c\right| = \left|4 - 3\right| \ne 13$

but \left|a - c\right| = \left|16 - 3\right| = 13$</p><p>Hence,${R}_{2}$is also not transitive.</p><p>Hence neither${R}_{1}$nor${R}_{2} is an equivalence relation