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

The Correct Option is B

To determine whether the relations \( R_1 \) and \( R_2 \) are equivalence relations, we need to check the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

Examination of \( R_1 = \{(a, b) \in \mathbb{N} \times \mathbb{N} : |a - b| \leq 13\} \)

• Reflexivity: For any natural number \( a \), \( |a - a| = 0 \). Since \( 0 \leq 13 \), \( (a, a) \in R_1 \). Hence, \( R_1 \) is reflexive.
• Symmetry: If \( (a, b) \in R_1 \), then \( |a - b| \leq 13 \). This implies that \( |b - a| = |a - b| \leq 13 \), so \( (b, a) \in R_1 \). Therefore, \( R_1 \) is symmetric.
• Transitivity: Suppose \( (a, b) \in R_1 \) and \( (b, c) \in R_1 \). Then \( |a - b| \leq 13 \) and \( |b - c| \leq 13 \). For \( R_1 \) to be transitive, \( |a - c| \leq 13 \) must hold. However, consider the case where \( a = 0 \), \( b = 13 \), and \( c = 26 \). Here, \( |a-b| = 13 \), \( |b-c| = 13 \), but \( |a-c| = 26 \), which is not less than or equal to 13. Thus, \( R_1 \) is not transitive.

Since \( R_1 \) is not transitive, it is not an equivalence relation.

Examination of \( R_2 = \{(a, b) \in \mathbb{N} \times \mathbb{N} : |a - b| \neq 13\} \)

• Reflexivity: For any natural number \( a \), \( |a - a| = 0 \neq 13 \). Hence, \( (a, a) \in R_2 \). \( R_2 \) is reflexive.
• Symmetry: If \( (a, b) \in R_2 \), then \( |a - b| \neq 13 \). This implies that \( |b - a| = |a - b| \neq 13 \), so \( (b, a) \in R_2 \). Therefore, \( R_2 \) is symmetric.
• Transitivity: Suppose \( (a, b) \in R_2 \) and \( (b, c) \in R_2 \). Then \( |a - b| \neq 13 \) and \( |b - c| \neq 13 \). For transitivity, we need \( |a - c| \neq 13 \). However, consider \( a = 0 \), \( b = 1 \), and \( c = 14 \). Here, \( |a - b| = 1 \neq 13 \) and \( |b - c| = 13 \neq 13 \), but \( |a - c| = 14 - 0 = 14 - 13 = 1 \neq 13 \) contradicts transitivity, showing \( R_2 \) is not transitive as well.

Since \( R_2 \) is not transitive, it is also not an equivalence relation.

Therefore, the correct answer is: Neither \( R_1 \) nor \( R_2 \) is an equivalence relation.