Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A = \{1, 2, 3, \ldots, 13, 14\}$ defined as $R = \{(x, y) : 3x - y = 0\}$.

(N/A) Given the set $A = \{1, 2, 3, \ldots, 13, 14\}$ and the relation $R = \{(x, y) : 3x - y = 0\}$.
We can rewrite the condition as $y = 3x$. For $x, y \in A$,the ordered pairs $(x, y)$ are:
$R = \{(1, 3), (2, 6), (3, 9), (4, 12)\}$.
$1$. Reflexive: $A$ relation $R$ is reflexive if $(a, a) \in R$ for all $a \in A$. Here,$(1, 1) \notin R$ because $3(1) - 1 = 2 \neq 0$. Thus,$R$ is not reflexive.
$2$. Symmetric: $A$ relation $R$ is symmetric if $(a, b) \in R \implies (b, a) \in R$. Here,$(1, 3) \in R$,but $(3, 1) \notin R$ because $3(3) - 1 = 8 \neq 0$. Thus,$R$ is not symmetric.
$3$. Transitive: $A$ relation $R$ is transitive if $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$. Here,$(1, 3) \in R$ and $(3, 9) \in R$,but $(1, 9) \notin R$ because $3(1) - 9 = -6 \neq 0$. Thus,$R$ is not transitive.
Conclusion: The relation $R$ is neither reflexive,nor symmetric,nor transitive.