Question

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

$\text{ (a, b) }R(c,d)\Rightarrow ad(b-c)=bc(a-d)$
Symmetric:
$(c,d)R(a,b)\Rightarrow cb(d-a)=da(c-b)\Rightarrow$
Symmetric Reflexive:
(a, b) $R(a,b)\Rightarrow ab(b-a)\ne ba(a-b)\Rightarrow$
Not reflexive
Transitive: $(2,3)R(3,2)$ and $(3,2)R(5,30)$ but
$((2,3),(5,30))\notin R\Rightarrow \text{ Not transitive }$