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 nature of the relation \( R \) defined on the cartesian product \( A \times B \), we are given that \((a, b)R(c, d)\) if and only if \(3ad - 7bc\) is an even integer. Let's analyze whether this relation is reflexive, symmetric, and transitive:

1. Reflexivity:
   • For reflexivity, each ordered pair \((a, b)\) with itself should satisfy the condition \((a, b)R(a, b)\).
   • Substituting \((a, b) = (a, b)\) into the relation condition, we have \(3ab - 7ab = -4ab\).
   • Clearly, \(-4ab\) is an even integer (since any integer multiplied by an even number is even).
   • Therefore, the relation \( R \) is reflexive on \( A \times B \).
2. Symmetry:
   • For symmetry, if \((a, b)R(c, d)\) then \((c, d)R(a, b)\) should also hold.
   • If \((a, b)R(c, d)\), we have \(3ad - 7bc\) is even.
   • We must check if \(3dc - 7ba\) is also even.
   • Since integer addition/subtraction preserves parity, \((3ad - 7bc) \equiv_2 (3dc - 7ba)\) implies symmetry.
   • Therefore, the relation \( R \) is symmetric.
3. Transitivity:
   • For transitivity, if \((a, b)R(c, d)\) and \((c, d)R(e, f)\), then \((a, b)R(e, f)\) should hold as well.
   • Given \((3ad - 7bc)\) and \((3cf - 7de)\) are even, nothing guarantees \((3af - 7be)\) is even without additional conditions.
   • Thus, the relation \( R \) is not transitive in general.

Based on the analysis, the relation \( R \) is reflexive and symmetric but not transitive. Thus, the correct answer is:

• Reflexive and symmetric but not transitive.

Answer 2

The relation \( R \) is defined on the Cartesian product \( A \times B \) such that \((a, b)R(c, d)\) if and only if \(3ad - 7bc\) is an even integer, where \(a, c \in A\) and \(b, d \in B\). Reflexivity: For the relation \( R \) to be reflexive, \((a, b)R(a, b)\) must hold for all \((a, b) \in A \times B\). We check: \[ 3ab - 7ba = -4ab, \]

which is always even since \(-4ab\) is a product of an integer and an even number. Therefore, \( R \) is reflexive. Symmetry: For \( R \) to be symmetric, if \((a, b)R(c, d)\) holds, then \((c, d)R(a, b)\) must also hold. Given that \(3ad - 7bc\) is even, consider swapping \((a, b)\) and \((c, d)\): \[ (c, d)R(a, b) \implies 3bc - 7ad. \] 

Since the difference between even numbers remains even, \( R \) is symmetric. Transitivity: For \( R \) to be transitive, if \((a, b)R(c, d)\) and \((c, d)R(e, f)\) hold, then \((a, b)R(e, f)\) must also hold. Consider specific elements: - Let \((3, 4)R(6, 4)\) hold, satisfying the condition as \(3 \cdot 6 \cdot 4 - 7 \cdot 4 \cdot 4\) is even. - Let \((6, 4)R(3, 1)\) hold, as \(3 \cdot 6 \cdot 1 - 7 \cdot 4 \cdot 1\) is also even. - However, \((3, 4)R(3, 1)\) does not satisfy the condition, as \(3 \cdot 3 \cdot 1 - 7 \cdot 4 \cdot 1\) is not necessarily even. Therefore, \( R \) is not transitive. Based on the above, the relation \( R \) is: reflexive and symmetric but not transitive.