Let R be a relation on N × N defined by (a, b) R (c, d) if and only if a d(b-c)=b c(a-d) . Then R i

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1.Relation and Function

hard

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

A

symmetric but neither reflexive nor transitive

B

transitive but neither reflexive nor symmetric

C

reflexive and symmetric but not transitive

D

symmetric and transitive but not reflexive

(JEE MAIN-2023)

Solution

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

Symmetric:

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

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

Standard 12

Mathematics

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(JEE MAIN-2023)