Question

Let R1 = {(a, b) ∈ N × N : |a - b| ≤ 13} and R2 = {(a, b) ∈ N × N : |a - b| ≠ 13}. Then on N:

• A. Both R1 and R2 are equivalence relations
• B. Neither R1 nor R2 is an equivalence relation
• C. R1 is an equivalence relation but R2 is not
• D. R2 is an equivalence relation but R1 is not

Answer 1

Answer: (B)

Neither R1 nor R2 is an equivalence relation

Answer 2

The correct answer is (B) : Neither R1 nor R2 is an equivalence relation
R1 = {(a, b) ∈ N × N : |a – b| ≤ 13} and
R2 = {(a, b) ∈ N × N : |a – b| ≠ 13}
In R1: ∵ |2 – 11| = 9 ≤ 13
∴ (2, 11) ∈ R1 and (11, 19) ∈ R1 but
(2, 19) ∉ R1
∴ R1 is not transitive
Hence R1 is not equivalence
In R2 : (13, 3) ∈ R2 and (3, 26) ∈ R2 but
(13, 26) ∉ R2 (∵ |13 – 26| = 13)
∴ R2 is not transitive
Hence R2 is not equivalence.