Let $R$ be a relation on $Z \times Z$ defined by $(a, b)R(c, d)$ if and only if $ad - bc$ is divisible by $5$. Then $R$ is
(A)
Reflexive and transitive but not symmetric
(B)
Reflexive and symmetric but not transitive
(C)
Reflexive but neither symmetric nor transitive
(D)
Reflexive, symmetric and transitive
MEDIUM
Correct Answer:B
Explanation
Reflexive:
Since $a*b - a*b = 0$ is divisible by $5$, $(a, b)R(a, b)$ holds.
Symmetric:
If $(a, b)R(c, d)$, then $ad - bc$ is divisible by $5$. Thus, $bc - ad$ is divisible by $5$, so $(c, d)R(a, b)$ holds.
Not Transitive:
Consider $(3, 1)R(10, 5)$ because $3*5 - 1*10 = 5$ is divisible by $5$.
Consider $(10, 5)R(1, 1)$ because $10*1 - 5*1 = 5$ is divisible by $5$.
But $(3, 1)$ is not related to $(1, 1)$ because $3*1 - 1*1 = 2$ is not divisible by $5$.
Hence, $R$ is reflexive and symmetric but not transitive.