Question

Relation $R$ on the set $A = \\{1, 2, 3, \ldots, 13, 14\\}$ defined as $R = \\{(x, y) : 3x - y = 0\\}$ is:

• A. Reflexive, symmetric and transitive
• B. Reflexive and transitive but not symmetric
• C. Neither reflexive nor symmetric but transitive
• D. Neither reflexive nor symmetric nor transitive

Answer 1

Answer: (C)

Neither reflexive nor symmetric but transitive

Answer 2

The given relation is $R = \\{(x, y) : 3x - y = 0\\}$.

• Reflexivity: For $R$ to be reflexive, $(x, x)$ must satisfy $3x - x = 0$ for all $x \in A$. However, $3x - x = 2x \neq 0$ for $x \neq 0$.
  Hence, $R$ is not reflexive.
• Symmetry: For $R$ to be symmetric, if $(x, y) \in R$, then $(y, x)$ must also belong to $R$.
  Check: $3x - y = 0 \implies 3y - x \neq 0 \text{ for } x \neq y.$ Hence, $R$ is not symmetric.
• Transitivity: For $R$ to be transitive, if $(x, y) \in R$ and $(y, z) \in R$, then $(x, z)$ must also belong to $R$.
  Check: $3x - y = 0 \text{ and } 3y - z = 0 \implies 3(3x) - z = 0 \implies 9x - z = 0,$ which satisfies the condition. Hence, $R$ is transitive.

Thus, the relation $R$ is neither reflexive nor symmetric but transitive.