Question

Let A= {2, 3, 6, 8, 9, 11} and B = {1, 4, 5, 10, 15} Let R be a relation on A × B define by (a, b)R(c, d) if and only if 3ad – 7bc is an even integer. Then the relation R is

• A. reflexive but not symmetric.
• B. transitive but not symmetric.
• C. reflexive and symmetric but not transitive
• D. an equivalence relation.

Answer 1

The Correct Option is C

To determine the properties of the relation \( R \) on \( A \times B \), where \((a, b)R(c, d)\) if and only if \(3ad - 7bc\) is an even integer, we examine reflexivity, symmetry, and transitivity:

1. Reflexivity:
   • For reflexivity, we test if \((a, b)R(a, b)\) holds for all \((a, b) \in A \times B\).
   • The condition becomes \(3ab - 7ab = -4ab\).
   • Since \(-4ab\) is always an even integer, the relation \( R \) is reflexive.
2. Symmetry:
   • For symmetry, we check if \((a, b)R(c, d)\) implies \((c, d)R(a, b)\).
   • If \(3ad - 7bc\) is even, we need to verify if \(3dc - 7ba\) is also even.
   • The parity of \(3ad - 7bc\) and \(3dc - 7ba\) are related. The statement \( (3ad - 7bc) \pmod 2 = 0 \) implies \( (3dc - 7ba) \pmod 2 = 0 \) because \(3ad - 7bc \equiv 3dc - 7ba \pmod 2\).
   • Therefore, the relation \( R \) is symmetric.
3. Transitivity:
   • For transitivity, we check if \((a, b)R(c, d)\) and \((c, d)R(e, f)\) implies \((a, b)R(e, f)\).
   • Given that \(3ad - 7bc\) and \(3cf - 7de\) are even, there is no guarantee that \(3af - 7be\) is even.
   • Thus, the relation \( R \) is not generally transitive.

In conclusion, the relation \( R \) is reflexive and symmetric, but not transitive. The classification is:

• Reflexive and symmetric but not transitive.