Question

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. transitive but neither reflexive nor symmetric
• B. symmetric but neither reflexive nor transitive
• C. symmetric and transitive but not reflexive
• D. reflexive and symmetric but not transitive

Answer 1

Answer: (B)

$\text { (a, b) } R(c, d) \Rightarrow a d(b-c)=b c(a-d)$
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
$((2,3),(5,30)) \notin R \Rightarrow \text { Not transitive }$