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

Relation: $$(a, b)R(c, d)$$ if $$3ad - 7bc$$ is even.

1. Reflexive: $$(a, b)R(a, b) \implies 3ab - 7ba = -4ab$$, which is always even. Yes.
2. Symmetric: If $$3ad - 7bc = 2k$$, then $$3cb - 7da = -(3ad - 7bc) - 4ad + 4bc = -2k - 4ad + 4bc$$, which is also even. Yes.
3. Transitive: Let $(2, 1)R(6, 5)$ because $$3(2)(5) - 7(1)(6) = 30 - 42 = -12$$ (even).

Let $$(6, 5)R(3, 10)$$ because $$3(6)(10) - 7(5)(3) = 180 - 105 = 75$$ (odd).

Check a valid pair: $$(a,b)R(c,d)$$ and $$(c,d)R(e,f)$$. Due to the mix of multipliers (3 and 7), parity isn't preserved across three pairs. No.

Result: Reflexive and symmetric but not transitive. (Option B)