Let R_1=a,b∈N×N:a-b≤13 and R_2=a,b∈N×N:a-b≠13 Then on N:

Q31 / 90 · JEE Main 2022 (28 Jun Shift 2) · Mathematics

Let $R_{1} = \{(a, b) \in N \times N : |a - b| \leq 13\}$ and $R_{2} = \{(a, b) \in N \times N : |a - b| \neq 13\}$ Then on $N$ :

To check if $R_{1}$ is transitive -

Let $a = 16, b = 4, c = 1$

Here $|a - b| = |16 - 4| \leq 13$ holds true

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

but $|a - c| = |16 - 1| ≰ 13$

Hence $R_{1}$ is not transitive

Similarly, to check transitivity of $R_{2}$

Let $a = 16, b = 4, c = 3$

Here $|a - b| = |16 - 4| \neq 13 and |b - c| = |4 - 3| \neq 13$

but $|a - c| = |16 - 3| = 13$

Hence, $R_{2}$ is also not transitive.

Hence neither $R_{1}$ nor $R_{2}$ is an equivalence relation

Hence, the correct answer is option: 'B'.