### Sets And Relations Question 5

#### Question 5 - 2024 (29 Jan Shift 1)

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

(1) Reflexive and symmetric but not transitive

(2) Reflexive but neither symmetric not transitive

(3) Reflexive, symmetric and transitive

(4) Reflexive and transitive but not symmetric

## Show Answer

#### Answer (1)

#### Solution

$(a, b) R(a, b)$ as $a b-a b=0$

Therefore reflexive

Let $(a, b) R(c, d) \Rightarrow a d-b c$ is divisible by $5 \Rightarrow bc-$ ad is divisible by $5 \Rightarrow(c, d) R(a, b)$

Therefore symmetric

Relation not transitive as $(3,1) R(10,5)$ and

$(10,5) R(1,1)$ but $(3,1)$ is not related to $(1,1)$