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

Step-by-step solution

Given relation R : a , b R c , d And a d - b c = 5 k where k ∈ Z Now, for reflexive R : a , b R a , b ⇒ a b - b a = 0 which is divisible by 5 So, the relation is reflexive, Now, for symmetric R : c , d R a , b ⇒ b c - a d = - 5 k which is divisible by 5 So, the relation is symmetric, Now for transitive relation, R : a , b R c , d R e , f Taking example a , b ≡ 1 , 2 , c , d ≡ 10 , 10 , e , f ≡ 4 , 4 Now, a d - b c = 10 - 20 = - 10 c f - e d = 40 - 40 = 0 But a f - b e = 8 - 4 = 4 which is not divisible by 5 , Hence