### Sets And Relations Question 5

#### Question 5 - 31 January - Shift 1

Let $R$ be a relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a, b) R$ $(c, d)$ if and only if $a d(b-c)=b c(a-d)$. Then $R$ is

(1) symmetric but neither reflexive nor transitive

(2) transitive but neither reflexive nor symmetric

(3) reflexive and symmetric but not transitive

(4) symmetric and transitive but not reflexive

## Show Answer

#### Answer: (1)

#### Solution:

#### Formula: Reflexive relation (iv), Symmetric relation (v), Transitive relation (vi)

(a, b) R $(c, d) \Rightarrow ad(b-c)=bc(a-d)$

Symmetric:

$(c, d) R(a, b) \Rightarrow cb(d-a)=da(c-b) \Rightarrow$ Symmetric

Reflexive:

$(a, b) R(a, b) \Rightarrow a b(b-a) \neq b a(a-b) \Rightarrow$ Not reflexive

Transitive: $(2,3) R(3,2)$ and $(3,2) R(5,30)$ but

$((2,3),(5,30)) \notin R \Rightarrow$ Not transitive