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

Step-by-step solution

To check if R 1 is transitive - Let a = 16 ,   b = 4 ,   c = 1 Here a - b = 16 - 4 ≤ 13 holds true now b - c = 4 − 1 ≤ 13 ⇒ 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 ≠ 13   and   b - c = 4 - 3 ≠ 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