Question

 Let \( R \) be a relation on \( \mathbb{Z} \times \mathbb{Z} \) defined by \((a, b) R (c, d)\) if and only if \(ad - bc\) is divisible by 5.
Then \( R \) is:

• A. Reflexive but neither symmetric nor transitive
• B. Reflexive and symmetric but not transitive
• C. Reflexive, symmetric and transitive
• D. Reflexive and transitive but not symmetric

Answer 1

The Correct Option is B

To ascertain the properties of the relation \( R \) on \( \mathbb{Z} \times \mathbb{Z} \), defined as \((a, b) R (c, d)\) if and only if \(ad - bc\) is divisible by 5, we examine reflexivity, symmetry, and transitivity.

1. Reflexivity:
   • For \( R \) to be reflexive, \((a, b) R (a, b)\) must hold for all \((a, b) \in \mathbb{Z} \times \mathbb{Z}\).
   • Substituting \((a, b)\) for \((c, d)\) yields \(ad - bc = ab - ba = 0\).
   • As 0 is divisible by 5, the relation is reflexive.
2. Symmetry:
   • For \( R \) to be symmetric, if \((a, b) R (c, d)\), then \((c, d) R (a, b)\) must be true.
   • If \(ad - bc\) is divisible by 5, then \(bc - ad = -(ad - bc)\) is also divisible by 5.
   • Therefore, the relation is symmetric.
3. Transitivity:
   • For \( R \) to be transitive, if \((a, b) R (c, d)\) and \((c, d) R (e, f)\), then \((a, b) R (e, f)\) must be true.
   • Let \(x = ad - bc\) and \(y = cf - de\), where both \(x\) and \(y\) are divisible by 5. For transitivity, \(af - be\) must also be divisible by 5.
   • However, the divisibility of \(x\) and \(y\) by 5 does not guarantee the divisibility of \(af - be\) by 5.
   • Consider the case where \(a = 1, b = 0, c = 0, d = 1, e = 1, f = 0\). Here, \(ad - bc = 1\) and \(cf - de = 1\), neither of which is divisible by 5. This example is invalid.
   • Let's re-evaluate with a valid example demonstrating non-transitivity. If \(ad - bc\) is divisible by 5 and \(cf - de\) is divisible by 5, it is not guaranteed that \(af - be\) is divisible by 5.
   • For instance, let \( (a,b) = (1,0) \), \( (c,d) = (5,0) \), and \( (e,f) = (0,1) \). Then \( ad - bc = 1(0) - 0(5) = 0 \), which is divisible by 5. Also, \( cf - de = 5(1) - 0(0) = 5 \), which is divisible by 5. However, \( af - be = 1(1) - 0(0) = 1 \), which is not divisible by 5.
   • Thus, the relation is not transitive.

Based on these analyses, the relation is reflexive and symmetric, but not transitive.