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)
symmetric and transitive but not reflexive
(B)
reflexive and symmetric but not transitive
(C)
transitive but neither reflexive nor symmetric
(D)
symmetric but neither reflexive nor transitive
MEDIUM
Correct Answer:D
Explanation
Given $(a, b) R (c, d) \implies ad(b - c) = bc(a - d)$.
Symmetric:
$(c, d) R (a, b) \implies cb(d - a) = da(c - b)$. This is symmetric.
Reflexive:
$(a, b) R (a, b) \implies ab(b - a) \neq ba(a - b)$. Thus, not reflexive.
Transitive:
$(2, 3) R (3, 2)$ and $(3, 2) R (5, 30)$ but $((2, 3), (5, 30)) \notin R$. Thus, not transitive.
Therefore, R is symmetric but neither reflexive nor transitive.