Question

** Let $R$ be a relation on $\mathbb{Z} \times \mathbb{Z}$ defined by **$(a, b) R (c, d)$ if and only if $ad - bc$ is divisible by 5.
Then $R$ is:

• A. Reflexive but neither symmetric nor transitive
• B. Reflexive and symmetric but not transitive
• C. Reflexive, symmetric and transitive
• D. Reflexive and transitive but not symmetric

Answer 1

Answer: (B)

Reflexive and symmetric but not transitive

Answer 2

Reflexive : for $(a, b) R (a, b)$
$⇒ ab – ab = 0$ is divisible by 5.
So $(a, b) R(a, b) ∀ a, b ∈ Z$
∴ R is reflexive Symmetric : For $(a, b) R(c, d)$
If $ad – bc$ is divisible by 5.
Then $bc – ad$ is also divisible by 5.
$⇒ (c, d) R(a, b) ∀ a, b, c, d ∈ Z$
∴ R is symmetric Transitive : If $(a, b) R(c, d)$
$⇒ ad – bc$ divisible by 5 and $(c, d) R (e, f)$
$⇒ cf – de$ divisible by 5
$ad – bc = 5k_1$ $k_1$ and $k_2$ are integers
$cf – de = 5k_2$
$afd – bcf = 5k_1f$
$bcf – bde = 5k_2b$
$afd – bde = 5(k_1f + k_2b)$
$d(af – be) = 5 (k_1f + k_2b)$
$⇒ af – be$ is not divisible by 5 for every a, b, c, d, e, f ∈ Z.
$∴$ R is not transitive
For e.g., take $a = 1, b = 2, c = 5, d = 5, e = 2, f = 2$

The correct option is (B): Reflexive and symmetric but not transitive