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)

symmetric but neither reflexive nor transitive

Answer 2

(a, b) R(c,d)⇒ad(b−c)=bc(a−d)
Symmetric:
(c,d)R(a,b)⇒cb(d−a)=da(c−b)⇒
Symmetric Reflexive:
(a, b) R(a,b)⇒ab(b−a)=ba(a−b)⇒
Not reflexive
Transitive: (2,3)R(3,2) and (3,2)R(5,30) but
((2,3),(5,30))∈/R⇒ Not transitive