Let R be a relation on defined by (a, b)?R?(c, d) if and only if is divisible by 5. Then R is

Question

Let R be a relation on $Z \times Z$ defined by (a, b) R (c, d) if and only if $a d - b c$ is divisible by 5. Then R is [2024]

Smart Achievers · Accepted Answer

(4)

$Reflexive : For (a, b) R (a, b)$

$\Rightarrow a b - a b = 0 is divisible by 5 .$

$So, (a, b) R (a, b)$ $\forall a, b \in Z$

$∴$ $R is reflexive .$

$Symmetric : For (a, b) R (c, d), if a d - b c is divisible by 5 .$

$Then, b c - a d is also divisible by 5 .$

$So, (c, d) R (a, b) \forall a, b, c, d \in Z$

$∴ R$ $is symmetric .$

$Transitive : For (a, b) R (c, d) \Rightarrow a d - b c is divisible by 5 and (c, d) R (e, f) \Rightarrow c f - d e is divisible by 5$

$L e t a d - b c = 5 k_{1} and c f - d e = 5 k_{2}, where k_{1} and k_{2} are integers .$

$∴ a d f - b c f = 5 k_{1} f$ ...(i)

$and c f b - d e b = 5 k_{2} b$ ...(ii)

$Solving (i) and (ii), we get$

$a d f - b c f + c f b - d e b = 5 k_{1} f + 5 k_{2} b$

$\Rightarrow a d f - d e b = 5 (k_{1} f + k_{2} b) \Rightarrow d (a f - b e) = 5 (k_{1} f + k_{2} b)$

$a f - b e is not divisible by 5 for \forall a, b, c, d, e, f \in Z$

$So, R is not transitive .$

$∴$ $R is Reflexive and symmetric but not transitive .$

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