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
(A)
reflexive but not symmetric.
(B)
an equivalence relation.
(C)
reflexive and symmetric but not transitive.
(D)
transitive but not symmetric.
MEDIUM
Correct Answer:C
Explanation
Given that $(a, b)R(c, d) \Rightarrow 3ad - 7bc \in \text{even}$.
For reflexive, we need to check if $(a, b)R(a, b)$.
$(a, b)R(a, b) \Rightarrow 3ab - 7ba = -4ab \in \text{even}$. This is true since the product $ab$ will be an integer, and $-4$ times an integer is even.
Thus, $R$ is reflexive.
For symmetric, if $(a, b)R(c, d)$, then $(c, d)R(a, b)$ must also be true.
$(a, b)R(c, d) \Rightarrow 3ad - 7bc = 2m$ for some integer $m$.
$(c, d)R(a, b) \Rightarrow 3cb - 7ad$. We want to show this is even.
$3cb - 7ad = -7ad + 3cb = -(7ad - 3cb) = -(7ad - 3cb + 3ad - 3ad - 7bc + 7bc) = -(10ad - 10bc - (3ad - 7bc)) = -10(ad - bc) + (3ad-7bc)$