Let R be a relation on Z × Z defined by ( a, b ) R ( c, d ) if and...

Given relation $R: \left(a, b\right)R \left(c, d\right)</p><p>And$ad - bc = 5k \left\{\text{where}k\in Z\right\}

Now, for reflexive $R: \left(a, b\right)$R \left(a, b\right)$</p><p>$⇒ ab - ba = 0$which is divisible by$5$</p><p>So, the relation is reflexive,$

Now, for symmetric $R: \left(c, d\right)$R \left(a, b\right)$</p><p>$⇒ bc - ad = - 5k$which is divisible by$5$</p><p>So, the relation is symmetric,$

Now for transitive relation, $R: \left(a, b\right)$R \left(c, d\right)$R \left(e, f\right)$

Taking example \left(a, b\right) ≡ \left(1, 2\right), \left(c, d\right) ≡ \left(10, 10\right), \left(e, f\right)$≡ \left(4, 4\right)$

Now, $ad - bc = 10 - 20 = - 10$

$cf - ed = 40 - 40 = 0$

But $af - be = 8 - 4 = 4$ which is not divisible by $5,$

Hence, the relation is reflexive, symmetric but not transitive.